9

Often a graph is the best way to represent a function because it conveys so much

information at a glance. Shown is a graph of the

vertical ground acceleration created by the 2011

earthquake near Tohoku, Japan. The earthquake

had a magnitude of 9.0 on the Richter scale and was

so powerful that it moved northern Japan 8 feet closer

to North America.

Functions and Models

The fundamenTal objecTs ThaT we deal with in calculus are functions. This chapter prepares the way for calculus by discussing the basic ideas concerning functions, their graphs, and ways of transforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of realworld phenomena.

1

Pictura Collectus/Alamy

Seismological Society of America

1000

_1000

2000

_2000

500 100 150 200

(cm/s@)

time

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

10 Chapter 1 Functions and Models

Functions arise whenever one quantity depends on another. Consider the following four situations.

A. The area A of a circle depends on the radius r of the circle. The rule that connects r and A is given by the equation A − �r 2. With each positive number r there is associated one value of A, and we say that A is a function of r.

B. The human population of the world P depends on the time t. The table gives estimates of the world population Pstd at time t, for certain years. For instance,

Ps1950d < 2,560,000,000

But for each value of the time t there is a corresponding value of P, and we say that P is a function of t.

C. The cost C of mailing an envelope depends on its weight w. Although there is no simple formula that connects w and C, the post office has a rule for determining C when w is known.

D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a.

{cm/s@}

(seconds)5

50

10 15 20 25

a

t

100

30

_50

Calif. Dept. of Mines and Geology

Each of these examples describes a rule whereby, given a number (r, t, w, or t), another number (A, P, C, or a) is assigned. In each case we say that the second number is a function of the first number.

A function f is a rule that assigns to each element x in a set D exactly one element, called f sxd, in a set E.

We usually consider functions for which the sets D and E are sets of real numbers. The set D is called the domain of the function. The number f sxd is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f sxd as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable.

YearPopulation (millions)

1900 16501910 17501920 18601930 20701940 23001950 25601960 30401970 37101980 44501990 52802000 60802010 6870

FIGURE 1Vertical ground acceleration

during the Northridge earthquake

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

SeCtion 1.1 Four Ways to Represent a Function 11

It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f sxd according to the rule of the function. Thus we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs.

The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function. You press the key labeled s (or sx ) and enter the input x. If x , 0, then x is not in the domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x > 0, then an approximation to sx will appear in the display. Thus the sx key on your calculator is not quite the same as the exact mathematical function f defined by f sxd − sx .

Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow connects an element of D to an element of E. The arrow indicates that f sxd is associated with x, f sad is associated with a, and so on.

The most common method for visualizing a function is its graph. If f is a function with domain D, then its graph is the set of ordered pairs

hsx, f sxdd | x [ Dj

(Notice that these are inputoutput pairs.) In other words, the graph of f consists of all points sx, yd in the coordinate plane such that y − f sxd and x is in the domain of f.

The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Since the ycoordinate of any point sx, yd on the graph is y − f sxd, we can read the value of f sxd from the graph as being the height of the graph above the point x (see Figure 4). The graph of f also allows us to picture the domain of f on the xaxis and its range on the yaxis as in Figure 5.

y � ƒ(x)

domain

range

{x, ƒ}

ƒ

f(1)f(2)

0 1 2 x xx

y y

example 1� The graph of a function f is shown in Figure 6.(a) Find the values of f s1d and f s5d.(b) What are the domain and range of f ?

Solution(a) We see from Figure 6 that the point s1, 3d lies on the graph of f, so the value of f at 1 is f s1d − 3. (In other words, the point on the graph that lies above x − 1 is 3 units above the xaxis.)

When x − 5, the graph lies about 0.7 units below the xaxis, so we estimate that f s5d < 20.7.

(b) We see that f sxd is defined when 0 < x < 7, so the domain of f is the closed interval f0, 7g. Notice that f takes on all values from 22 to 4, so the range of f is

hy | 22 < y < 4j − f22, 4g ■

x(input)

ƒ(output)

f

FIGURE 2Machine diagram for a function f

fD E

ƒ

f(a)a

x

FIGURE 3Arrow diagram for f

FIGURE 4 FIGURE 5

x

y

1

1

FIGURE 6

The notation for intervals is given in Appendix A.

Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).

Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

12 Chapter 1 Functions and Models

example 2� Sketch the graph and find the domain and range of each function.(a) fsxd − 2x 2 1 (b) tsxd − x 2

Solution(a) The equation of the graph is y − 2x 2 1, and we recognize this as being the equation of a line with slope 2 and yintercept 21. (Recall the slopeintercept form of the equation of a line: y − mx 1 b. See Appendix B.) This enables us to sketch a portion of the graph of f in Figure 7. The expression 2x 2 1 is defined for all real numbers, so the domain of f is the set of all real numbers, which we denote by R. The graph shows that the range is also R.

(b) Since ts2d − 22 − 4 and ts21d − s21d2 − 1, we could plot the points s2, 4d and s21, 1d, together with a few other points on the graph, and join them to produce the graph (Figure 8). The equation of the graph is y − x 2, which represents a parabola (see Appendix C). The domain of t is R. The range of t consists of all values of tsxd, that is, all numbers of the form x 2. But x 2 > 0 for all numbers x and any positive number y is a square. So the range of t is hy | y > 0j − f0, `d. This can also be seen from Figure 8. ■

example 3� If f sxd − 2x 2 2 5x 1 1 and h ± 0, evaluate f sa 1 hd 2 f sad

h.

Solution We first evaluate f sa 1 hd by replacing x by a 1 h in the expression for f sxd:

f sa 1 hd − 2sa 1 hd2 2 5sa 1 hd 1 1

− 2sa2 1 2ah 1 h2d 2 5sa 1 hd 1 1

− 2a2 1 4ah 1 2h2 2 5a 2 5h 1 1

Then we substitute into the given expression and simplify:

f sa 1 hd 2 f sadh

−s2a2 1 4ah 1 2h2 2 5a 2 5h 1 1d 2 s2a2 2 5a 1 1d

h

−2a2 1 4ah 1 2h2 2 5a 2 5h 1 1 2 2a2 1 5a 2 1

h

−4ah 1 2h2 2 5h

h− 4a 1 2h 2 5

■

representations of FunctionsThere are four possible ways to represent a function:

● verbally (by a description in words)● numerically (by a table of values)● visually (by a graph)● algebraically (by an explicit formula)

If a single function can be represented in all four ways, it’s often useful to go from one representation to another to gain additional insight into the function. (In Example 2, for instance, we started with algebraic formulas and then obtained the graphs.) But certain functions are described more naturally by one method than by another. With this in mind, let’s reexamine the four situations that we considered at the beginning of this section.

x

y=2x-1

0-1

y

12

FIGURE 7

(_1,1)

(2,4)

y

1

x1

y=≈

FIGURE 8

The expression

f sa 1 hd 2 f sadh

in Example 3 is called a difference quotient and occurs frequently in calculus. As we will see in Chapter 2, it represents the average rate of change of f sxd between x − a and x − a 1 h.

SeCtion 1.1 Four Ways to Represent a Function 13

A. The most useful representation of the area of a circle as a function of its radius is probably the algebraic formula Asrd − �r 2, though it is possible to compile a table of values or to sketch a graph (half a parabola). Because a circle has to have a positive radius, the domain is hr | r . 0j − s0, `d, and the range is also s0, `d.

B. We are given a description of the function in words: Pstd is the human population of the world at time t. Let’s measure t so that t − 0 corresponds to the year 1900. The table of values of world population provides a convenient representation of this function. If we plot these values, we get the graph (called a scatter plot) in Figure 9. It too is a useful representation; the graph allows us to absorb all the data at once. What about a formula? Of course, it’s impossible to devise an explicit formula that gives the exact human population Pstd at any time t. But it is possible to find an expression for a function that approximates Pstd. In fact, using methods explained in Section 1.2, we obtain the approximation

Pstd < f std − s1.43653 3 109d ∙ s1.01395dt

Figure 10 shows that it is a reasonably good “fit.” The function f is called a mathe-matical model for population growth. In other words, it is a function with an explicit formula that approximates the behavior of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an explicit formula is not necessary.

5x10' 5x10'

P

t20 40 60 80 100 120 20 40 60Years since 1900Years since 1900

80 100 120

P

t0 0

FIGURE 9 FIGURE 10

The function P is typical of the functions that arise whenever we attempt to apply calculus to the real world. We start with a verbal description of a function. Then we may be able to construct a table of values of the function, perhaps from instrument readings in a scientific experiment. Even though we don’t have complete knowledge of the values of the function, we will see throughout the book that it is still possible to perform the operations of calculus on such a function.

C. Again the function is described in words: Let Cswd be the cost of mailing a large envelope with weight w. The rule that the US Postal Service used as of 2015 is as follows: The cost is 98 cents for up to 1 oz, plus 21 cents for each additional ounce (or less) up to 13 oz. The table of values shown in the margin is the most convenient representation for this function, though it is possible to sketch a graph (see Example 10).

D. The graph shown in Figure 1 is the most natural representation of the vertical acceleration function astd. It’s true that a table of values could be compiled, and it is even possible to devise an approximate formula. But everything a geologist needs to

t (years

since 1900)Population (millions)

0 165010 175020 186030 207040 230050 256060 304070 371080 445090 5280

100 6080110 6870

A function defined by a table of values is called a tabular function.

w (ounces) Cswd (dollars)

0 , w < 1 0.98

1 , w < 2 1.19

2 , w < 3 1.40

3 , w < 4 1.61

4 , w < 5 1.82∙ ∙∙ ∙∙ ∙

14 Chapter 1 Functions and Models

PS In setting up applied functions as in Example 5, it may be useful to review the principles of problem solving as discussed on page 71, particularly Step 1: Understand the Problem.

know— amplitudes and patterns — can be seen easily from the graph. (The same is true for the patterns seen in electrocardiograms of heart patients and polygraphs for liedetection.)

In the next example we sketch the graph of a function that is defined verbally.

example 4 When you turn on a hotwater faucet, the temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on.

Solution The initial temperature of the running water is close to room temperature because the water has been sitting in the pipes. When the water from the hotwater tank starts flowing from the faucet, T increases quickly. In the next phase, T is constant at the tempera ture of the heated water in the tank. When the tank is drained, T decreases to the temperature of the water supply. This enables us to make the rough sketch of T as a function of t in Figure 11. ■

In the following example we start with a verbal description of a function in a physical situation and obtain an explicit algebraic formula. The ability to do this is a useful skill in solving calculus problems that ask for the maximum or minimum values of quantities.

example 5� A rectangular storage container with an open top has a volume of 10 m3. The length of its base is twice its width. Material for the base costs $10 per square meter; material for the sides costs $6 per square meter. Express the cost of materials as a function of the width of the base.

Solution We draw a diagram as in Figure 12 and introduce notation by letting w and 2w be the width and length of the base, respectively, and h be the height.

The area of the base is s2wdw − 2w2, so the cost, in dollars, of the material for the base is 10s2w2 d. Two of the sides have area wh and the other two have area 2wh, so the cost of the material for the sides is 6f2swhd 1 2s2whdg. The total cost is therefore

C − 10s2w2 d 1 6f2swhd 1 2s2whdg − 20w2 1 36wh

To express C as a function of w alone, we need to eliminate h and we do so by using the fact that the volume is 10 m3. Thus

ws2wdh − 10

which gives h −10

2w2 −5

w2

Substituting this into the expression for C, we have

C − 20w2 1 36wS 5

w2D − 20w2 1180

w

Therefore the equation

Cswd − 20w2 1180

w w . 0

expresses C as a function of w. ■

example 6� Find the domain of each function.

(a) f sxd − sx 1 2 (b) tsxd −1

x 2 2 x

t

T

FIGURE 11

w

2w

h

FIGURE 12

SeCtion 1.1 Four Ways to Represent a Function 15

Solution(a) Because the square root of a negative number is not defined (as a real number), the domain of f consists of all values of x such that x 1 2 > 0. This is equivalent to x > 22, so the domain is the interval f22, `d.(b) Since

tsxd −1

x 2 2 x−

1

xsx 2 1d

and division by 0 is not allowed, we see that tsxd is not defined when x − 0 or x − 1. Thus the domain of t is

hx | x ± 0, x ± 1j

which could also be written in interval notation as

s2`, 0d ø s0, 1d ø s1, `d ■

The graph of a function is a curve in the xyplane. But the question arises: Which curves in the xyplane are graphs of functions? This is answered by the following test.

The Vertical Line Test A curve in the xyplane is the graph of a function of x if and only if no vertical line intersects the curve more than once.

The reason for the truth of the Vertical Line Test can be seen in Figure 13. If each vertical line x − a intersects a curve only once, at sa, bd, then exactly one function value is defined by f sad − b. But if a line x − a intersects the curve twice, at sa, bd and sa, cd, then the curve can’t represent a function because a function can’t assign two different values to a.

For example, the parabola x − y 2 2 2 shown in Figure 14(a) is not the graph of a function of x because, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of x. Notice that the equation x − y 2 2 2 implies y 2 − x 1 2, so y − 6sx 1 2 . Thus the upper and lower halves of the parabola are the graphs of the functions f sxd − sx 1 2 [from Example 6(a)] and tsxd − 2sx 1 2 . [See Figures 14(b) and (c).]

We observe that if we reverse the roles of x and y, then the equation x − hsyd − y 2 2 2 does define x as a function of y (with y as the independent variable and x as the dependent variable) and the parabola now appears as the graph of the function h.

(b) y=œ„„„„x+2

_2 0 x

y

(_2, 0)

(a) x=¥-2

0 x

y

(c) y=_œ„„„„x+2

_20

y

x

piecewise Defined FunctionsThe functions in the following four examples are defined by different formulas in different parts of their domains. Such functions are called piecewise defined functions.

a

x=a

(a, b)

a

(a, c)

(a, b)

x=a

0 x

y

x

y

(a) This curve represents a function.

(b) This curve doesn’t represent a function.

FIGURE 13

FIGURE 14

domain conventionIf a function is given by a formula and the domain is not stated explicitly, the convention is that the domain is the set of all numbers for which the formula makes sense and defines a real number.

16 Chapter 1 Functions and Models

example 7 A function f is defined by

f sxd − H1 2 x

x 2

if x < 21

if x . 21

Evaluate f s22d, f s21d, and f s0d and sketch the graph.

Solution Remember that a function is a rule. For this particular function the rule is the following: First look at the value of the input x. If it happens that x < 21, then the value of f sxd is 1 2 x. On the other hand, if x . 21, then the value of f sxd is x 2.

Since 22 < 21, we have f s22d − 1 2 s22d − 3.

Since 21 < 21, we have f s21d − 1 2 s21d − 2.

Since 0 . 21, we have f s0d − 02 − 0.

How do we draw the graph of f ? We observe that if x < 21, then f sxd − 1 2 x, so the part of the graph of f that lies to the left of the vertical line x − 21 must coincide with the line y − 1 2 x, which has slope 21 and yintercept 1. If x . 21, then f sxd − x 2, so the part of the graph of f that lies to the right of the line x − 21 must coincide with the graph of y − x 2, which is a parabola. This enables us to sketch the graph in Figure 15. The solid dot indicates that the point s21, 2d is included on the graph; the open dot indicates that the point s21, 1d is excluded from the graph. ■

The next example of a piecewise defined function is the absolute value function. Recall that the absolute value of a number a, denoted by | a |, is the distance from a to 0 on the real number line. Distances are always positive or 0, so we have

| a | > 0 for every number a

For example,

| 3 | − 3 | 23 | − 3 | 0 | − 0 | s2 2 1 | − s2 2 1 | 3 2 � | − � 2 3

In general, we have

| a | − a if a > 0

| a | − 2a if a , 0

(Remember that if a is negative, then 2a is positive.)

example 8 Sketch the graph of the absolute value function f sxd − | x |.Solution From the preceding discussion we know that

| x | − Hx

2x

if x > 0

if x , 0

Using the same method as in Example 7, we see that the graph of f coincides with the line y − x to the right of the yaxis and coincides with the line y − 2x to the left of the yaxis (see Figure 16). ■

1

x

y

1_1 0

FIGURE 15

For a more extensive review of absolute values, see Appendix A.

x

y=| x |

y

FIGURE 16

SeCtion 1.1 Four Ways to Represent a Function 17

Pointslope form of the equation of a line:

y 2 y1 − msx 2 x1 d

See Appendix B.

example 9 Find a formula for the function f graphed in Figure 17.

Solution The line through s0, 0d and s1, 1d has slope m − 1 and yintercept b − 0, so its equation is y − x. Thus, for the part of the graph of f that joins s0, 0d to s1, 1d, we have

f sxd − x if 0 < x < 1

The line through s1, 1d and s2, 0d has slope m − 21, so its pointslope form is

y 2 0 − s21dsx 2 2d or y − 2 2 x

So we have f sxd − 2 2 x if 1 , x < 2

We also see that the graph of f coincides with the xaxis for x . 2. Putting this information together, we have the following threepiece formula for f :

f sxd − Hx

2 2 x

if 0 < x < 1

if 1 , x < 2

if x . 2 ■

example 1�0� In Example C at the beginning of this section we considered the cost Cswd of mailing a large envelope with weight w. In effect, this is a piecewise defined function because, from the table of values on page 13, we have

Cswd −

0.98

1.19

1.40

1.61

if 0 , w < 1

if 1 , w < 2

if 2 , w < 3

if 3 , w < 4 ∙ ∙ ∙

The graph is shown in Figure 18. You can see why functions similar to this one are called step functions—they jump from one value to the next. Such functions will be studied in Chapter 2. ■

SymmetryIf a function f satisfies f s2xd − f sxd for every number x in its domain, then f is called an even function. For instance, the function f sxd − x 2 is even because

f s2xd − s2xd2 − x 2 − f sxd

The geometric significance of an even function is that its graph is symmetric with respect to the yaxis (see Figure 19). This means that if we have plotted the graph of f for x > 0, we obtain the entire graph simply by reflecting this portion about the yaxis.

If f satisfies f s2xd − 2f sxd for every number x in its domain, then f is called an odd function. For example, the function f sxd − x 3 is odd because

f s2xd − s2xd3 − 2x 3 − 2f sxd

x

y

0 1

1

FIGURE 17

FIGURE 19 An even function

0 x_x

f(_x) ƒ

x

y

C

0.50

1.00

1.50

0 1 2 3 54 w

FIGURE 18

18 Chapter 1 Functions and Models

The graph of an odd function is symmetric about the origin (see Figure 20). If we already have the graph of f for x > 0, we can obtain the entire graph by rotating this portion through 1808 about the origin.

example 1�1� Determine whether each of the following functions is even, odd, or neither even nor odd.(a) f sxd − x 5 1 x (b) tsxd − 1 2 x 4 (c) hsxd − 2x 2 x 2

Solution(a) f s2xd − s2xd5 1 s2xd − s21d5x 5 1 s2xd

− 2x 5 2 x − 2sx 5 1 xd

− 2f sxd

Therefore f is an odd function.

(b) ts2xd − 1 2 s2xd4 − 1 2 x 4 − tsxdSo t is even.

(c) hs2xd − 2s2xd 2 s2xd2 − 22x 2 x 2

Since hs2xd ± hsxd and hs2xd ± 2hsxd, we conclude that h is neither even nor odd. ■

The graphs of the functions in Example 11 are shown in Figure 21. Notice that the graph of h is symmetric neither about the yaxis nor about the origin.

1

1 x

y

h1

1

y

x

g1

_1

1

y

x

f

_1

(a) (b) (c)

increasing and Decreasing FunctionsThe graph shown in Figure 22 rises from A to B, falls from B to C, and rises again from C to D. The function f is said to be increasing on the interval fa, bg, decreasing on fb, cg, and increasing again on fc, dg. Notice that if x1 and x2 are any two numbers between a and b with x1 , x2, then f sx1 d , f sx2 d. We use this as the defining property of an increasing function.

A

B

C

D

y=ƒ

f(x¡)

a

y

0 xx¡ x™ b c d

f(x™)

FIGURE 20 An odd function

0x

_x ƒx

y

FIGURE 21

FIGURE 22

SeCtion 1.1 Four Ways to Represent a Function 19

A function f is called increasing on an interval I if

f sx1 d , f sx2 d whenever x1 , x2 in I

It is called decreasing on I if

f sx1 d . f sx2 d whenever x1 , x2 in I

In the definition of an increasing function it is important to realize that the inequality f sx1 d , f sx2 d must be satisfied for every pair of numbers x1 and x2 in I with x1 , x2.

You can see from Figure 23 that the function f sxd − x 2 is decreasing on the interval s2`, 0g and increasing on the interval f0, `d.FIGURE 23

y

x

y=≈

1�. If f sxd − x 1 s2 2 x and tsud − u 1 s2 2 u , is it true that f − t?

2�. If

f sxd −x 2 2 x

x 2 1 and tsxd − x

is it true that f − t?

3�. The graph of a function f is given. (a) State the value of f s1d. (b) Estimate the value of f s21d. (c) For what values of x is f sxd − 1? (d) Estimate the value of x such that f sxd − 0. (e) State the domain and range of f. (f) On what interval is f increasing?

y

0 x1

1

4. The graphs of f and t are given.

g

x

y

f2

2

(a) State the values of f s24d and ts3d. (b) For what values of x is f sxd − tsxd?

(c) Estimate the solution of the equation f sxd − 21. (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of t.

5�. Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.

6�. In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.

7–1�0� Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.

7. 8. y

x0 1

1

y

x0

1

1

y

x0 1

1

y

x0 1

1

9. 1�0�.

1.1 ExErcisEs

20 Chapter 1 Functions and Models

11. Shown is a graph of the global average temperature T during the 20th century. Estimate the following.

(a) The global average temperature in 1950 (b) The year when the average temperature was 14.2°C (c) The year when the temperature was smallest? Largest? (d) The range of T

t

T (•C)

1900 1950 2000

13

14

Source: Adapted from Globe and Mail [Toronto], 5 Dec. 2009. Print.

12. Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to 2000.

(a) What is the range of the ring width function? (b) What does the graph tend to say about the temperature

of the earth? Does the graph reflect the volcanic erup-tions of the mid-19th century?

Rin

g w

idth

(m

m)

1.61.41.2

10.80.60.40.2

01500 1600 1700 1800 1900

Year

2000 t

R

Source: Adapted from G. Jacoby et al., “Mongolian Tree Rings and 20th-Century Warming,” Science 273 (1996): 771–73.

13. You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the tempera-ture of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.

14. Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?

100

20

A B Cy

15. The graph shows the power consumption for a day in Septem-ber in San Francisco. (P is measured in megawatts; t is mea-sured in hours starting at midnight.)

(a) What was the power consumption at 6 am? At 6 pm? (b) When was the power consumption the lowest? When was

it the highest? Do these times seem reasonable?

P

0 181512963 t21

400

600

800

200

Pacific Gas & Electric

16. Sketch a rough graph of the number of hours of daylight as a function of the time of year.

17. Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.

18. Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.

19. Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.

20. You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.

21. A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.

22. An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let xstd be the horizontal distance traveled and ystd be the altitude of the plane.

(a) Sketch a possible graph of xstd. (b) Sketch a possible graph of ystd.

SeCtion 1.1 Four Ways to Represent a Function 21

(c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity.

2�3�. Temperature readings T (in °F) were recorded every two hours from midnight to 2:00 pm in Atlanta on June 4, 2013. The time t was measured in hours from midnight.

t 0 2 4 6 8 10 12 14

T 74 69 68 66 70 78 82 86

(a) Use the readings to sketch a rough graph of T as a function of t.

(b) Use your graph to estimate the temperature at 9:00 am.

2�4. Researchers measured the blood alcohol concentration (BAC) of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC (in mgymL) of the eight men.

(a) Use the readings to sketch the graph of the BAC as a function of t.

(b) Use your graph to describe how the effect of alcohol varies with time.

t (hours) BAC t (hours) BAC

0 0 1.75 0.220.2 0.25 2.0 0.180.5 0.41 2.25 0.150.75 0.40 2.5 0.121.0 0.33 3.0 0.071.25 0.29 3.5 0.031.5 0.24 4.0 0.01

Source: Adapted from P. Wilkinson et al., “Pharmaco*kinetics of Ethanol after Oral Administration in the Fasting State,” Journal of Pharmaco*kinetics and Biopharmaceutics 5 (1977): 207–24.

2�5�. If f sxd − 3x 2 2 x 1 2, find f s2d, f s22d, f sad, f s2ad, f sa 1 1d, 2 f sad, f s2ad, f sa2d, [ f sad]2, and f sa 1 hd.

2�6�. A spherical balloon with radius r inches has volume Vsrd − 4

3 �r 3. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r 1 1 inches.

2�7–3�0� Evaluate the difference quotient for the given function. Simplify your answer.

2�7. f sxd − 4 1 3x 2 x 2, f s3 1 hd 2 f s3d

h

2�8. f sxd − x 3, f sa 1 hd 2 f sad

h

2�9. f sxd −1

x,

f sxd 2 f sadx 2 a

3�0�. f sxd −x 1 3

x 1 1,

f sxd 2 f s1dx 2 1

3�1�–3�7 Find the domain of the function.

3�1�. f sxd −x 1 4

x 2 2 9 3�2�. f sxd −

2x 3 2 5

x 2 1 x 2 6

3�3�. f std − s3 2t 2 1 3�4. tstd − s3 2 t 2 s2 1 t

3�5�. hsxd −1

s4 x 2 2 5x 3�6�. f sud −

u 1 1

1 11

u 1 1 3�7. Fspd − s2 2 sp

3�8. Find the domain and range and sketch the graph of the function hsxd − s4 2 x 2 .

3�9–40� Find the domain and sketch the graph of the function.

3�9. f sxd − 1.6x 2 2.4 40�. tstd −t 2 2 1

t 1 1

41�–44 Evaluate f s23d, f s0d, and f s2d for the piecewise defined function. Then sketch the graph of the function.

41�. f sxd − Hx 1 2

1 2 x

if x , 0

if x > 0

42�. f sxd − H3 2 12 x

2x 2 5

if x , 2

if x > 2

43�. f sxd − Hx 1 1

x 2

if x < 21

if x . 21

44. f sxd − H21

7 2 2x

if x < 1

if x . 1

45�–5�0� Sketch the graph of the function.

45�. f sxd − x 1 | x | 46�. f sxd − | x 1 2 | 47. tstd − |1 2 3t | 48. hstd − | t | 1 | t 1 1| 49. f sxd − H| x |

1

if | x | < 1

if | x | . 1 5�0�. tsxd − || x | 2 1|

5�1�–5�6� Find an expression for the function whose graph is the given curve.

5�1�. The line segment joining the points s1, 23d and s5, 7d

5�2�. The line segment joining the points s25, 10d and s7, 210d

5�3�. The bottom half of the parabola x 1 sy 2 1d2 − 0

5�4. The top half of the circle x 2 1 sy 2 2d2 − 4

22 Chapter 1 Functions and Models

5�5�. y

0 x

1

1

5�6�. y

0 x

1

1

5�7–6�1� Find a formula for the described function and state its domain.

5�7. A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.

5�8. A rectangle has area 16 m2. Express the perimeter of the rectangle as a function of the length of one of its sides.

5�9. Express the area of an equilateral triangle as a function of the length of a side.

6�0�. A closed rectangular box with volume 8 ft3 has length twice the width. Express the height of the box as a function of the width.

6�1�. An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base.

6�2�. A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.

x

6�3�. A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.

20

12x

x

x

x

x x

x x

6�4. A cell phone plan has a basic charge of $35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0 < x < 600.

6�5�. In a certain state the maximum speed permitted on freeways is 65 miyh and the minimum speed is 40 miyh. The fine for violating these limits is $15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed x and graph Fsxd for 0 < x < 100.

6�6�. An electricity company charges its customers a base rate of $10 a month, plus 6 cents per kilowatthour (kWh) for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount x of electricity used. Then graph the function E for 0 < x < 2000.

6�7. In a certain country, income tax is assessed as follows. There is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%.

(a) Sketch the graph of the tax rate R as a function of the income I.

(b) How much tax is assessed on an income of $14,000? On $26,000?

(c) Sketch the graph of the total assessed tax T as a function of the income I.

6�8. The functions in Example 10 and Exercise 67 are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.

6�9–70� Graphs of f and t are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.

6�9. y

x

f

g 70�. y

x

f

g

71�. (a) If the point s5, 3d is on the graph of an even function, what other point must also be on the graph?

(b) If the point s5, 3d is on the graph of an odd function, what other point must also be on the graph?

72�. A function f has domain f25, 5g and a portion of its graph is shown.

(a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd.

SeCtion 1.2 Mathematical Models: A Catalog of Essential Functions 23

x0

y

5_5

73�–78 Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.

73�. f sxd −x

x 2 1 1 74. f sxd −

x 2

x 4 1 1

75�. f sxd −x

x 1 1 76�. f sxd − x | x |

77. f sxd − 1 1 3x 2 2 x 4

78. f sxd − 1 1 3x 3 2 x 5

79. If f and t are both even functions, is f 1 t even? If f and t are both odd functions, is f 1 t odd? What if f is even and t is odd? Justify your answers.

80�. If f and t are both even functions, is the product ft even? If f and t are both odd functions, is ft odd? What if f is even and t is odd? Justify your answers.

A mathematical model is a mathematical description (often by means of a function or an equation) of a realworld phenomenon such as the size of a population, the demand for a product, the speed of a falling object, the concentration of a product in a chemical reaction, the life expectancy of a person at birth, or the cost of emission reductions. The purpose of the model is to understand the phenomenon and perhaps to make predictions about future behavior.

Figure 1 illustrates the process of mathematical modeling. Given a realworld problem, our first task is to formulate a mathematical model by identifying and naming the independent and dependent variables and making assumptions that simplify the phenomenon enough to make it mathematically tractable. We use our knowledge of the physical situation and our mathematical skills to obtain equations that relate the variables. In situations where there is no physical law to guide us, we may need to collect data (either from a library or the Internet or by conducting our own experiments) and examine the data in the form of a table in order to discern patterns. From this numeri cal representation of a function we may wish to obtain a graphical representation by plotting the data. The graph might even suggest a suitable algebraic formula in some cases.

Real-worldproblem

Mathematicalmodel

Real-worldpredictions

Mathematicalconclusions

Test

Formulate Solve Interpret

The second stage is to apply the mathematics that we know (such as the calculus that will be developed throughout this book) to the mathematical model that we have formulated in order to derive mathematical conclusions. Then, in the third stage, we take those mathematical conclusions and interpret them as information about the original realworld phenomenon by way of offering explanations or making predictions. The final step is to test our predictions by checking against new real data. If the predictions don’t compare well with reality, we need to refine our model or to formulate a new model and start the cycle again.

A mathematical model is never a completely accurate representation of a physical situation—it is an idealization. A good model simplifies reality enough to permit math

FIGURE 1The modeling process

24 Chapter 1 Functions and Models

ematical calculations but is accurate enough to provide valuable conclusions. It is important to realize the limitations of the model. In the end, Mother Nature has the final say.

There are many different types of functions that can be used to model relationships observed in the real world. In what follows, we discuss the behavior and graphs of these functions and give examples of situations appropriately modeled by such functions.

linear ModelsWhen we say that y is a linear function of x, we mean that the graph of the function is a line, so we can use the slopeintercept form of the equation of a line to write a formula for the function as

y − f sxd − mx 1 b

where m is the slope of the line and b is the yintercept.A characteristic feature of linear functions is that they grow at a constant rate. For

instance, Figure 2 shows a graph of the linear function f sxd − 3x 2 2 and a table of sample values. Notice that whenever x increases by 0.1, the value of f sxd increases by 0.3. So f sxd increases three times as fast as x. Thus the slope of the graph y − 3x 2 2, namely 3, can be interpreted as the rate of change of y with respect to x.

x

y

y=3x-2

_2

1

x f sxd − 3x 2 2

1.0 1.01.1 1.31.2 1.61.3 1.91.4 2.21.5 2.5

example 1� (a) As dry air moves upward, it expands and cools. If the ground temperature is 20°C and the temperature at a height of 1 km is 10°C, express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate.(b) Draw the graph of the function in part (a). What does the slope represent?(c) What is the temperature at a height of 2.5 km?

Solution(a) Because we are assuming that T is a linear function of h, we can write

T − mh 1 b

We are given that T − 20 when h − 0, so

20 − m ? 0 1 b − b

In other words, the yintercept is b − 20.We are also given that T − 10 when h − 1, so

10 − m ? 1 1 20

The slope of the line is therefore m − 10 2 20 − 210 and the required linear function is

T − 210h 1 20

The coordinate geometry of lines is reviewed in Appendix B.

FIGURE 2

Section 1.2 Mathematical Models: A Catalog of Essential Functions 25

(b) The graph is sketched in Figure 3. The slope is m − 2 10°Cykm, and this repre- sents the rate of change of temperature with respect to height.

(c) At a height of h − 2.5 km, the temperature is

T − 210s2.5d 1 20 − 2 5°C ■

If there is no physical law or principle to help us formulate a model, we construct an empirical model, which is based entirely on collected data. We seek a curve that “fits” the data in the sense that it captures the basic trend of the data points.

ExamplE 2 Table 1 lists the average carbon dioxide level in the atmosphere, mea-sured in parts per million at Mauna Loa Observatory from 1980 to 2012. Use the data in Table 1 to find a model for the carbon dioxide level.

SoLUtion We use the data in Table 1 to make the scatter plot in Figure 4, where t rep-resents time (in years) and C represents the CO2 level (in parts per million, ppm).

C (ppm)

340

350

360

370

380

390

400

1980 1985 t1990 1995 2000 2005 2010

FIGURE 4 Scatter plot for the average CO2 level

Notice that the data points appear to lie close to a straight line, so it’s natural to choose a linear model in this case. But there are many possible lines that approximate these data points, so which one should we use? One possibility is the line that passes through the first and last data points. The slope of this line is

393.8 2 338.7

2012 2 1980−

55.1

32− 1.721875 < 1.722

We write its equation as

C 2 338.7 − 1.722st 2 1980dor

1 C − 1.722t 2 3070.86

FIGURE 3

T=_10h+20

T

h0

10

20

1 3

YearCO2 level(in ppm) Year

CO2 level(in ppm)

1980 338.7 1998 366.51982 341.2 2000 369.41984 344.4 2002 373.21986 347.2 2004 377.51988 351.5 2006 381.91990 354.2 2008 385.61992 356.3 2010 389.91994 358.6 2012 393.81996 362.4

Table 1

26 Chapter 1 Functions and Models

Equation 1 gives one possible linear model for the carbon dioxide level; it is graphed in Figure 5.

C (ppm)

340

350

360

370

380

390

400

1980 1985 t1990 1995 2000 2005 2010

Notice that our model gives values higher than most of the actual CO2 levels. A better linear model is obtained by a procedure from statistics called linear regression. If we use a graphing calculator, we enter the data from Table 1 into the data editor and choose the linear regression command. (With Maple we use the fit[leastsquare] com-mand in the stats package; with Mathematica we use the Fit command.) The machine gives the slope and y-intercept of the regression line as

m − 1.71262 b − 23054.14

So our least squares model for the CO2 level is

2 C − 1.71262t 2 3054.14

In Figure 6 we graph the regression line as well as the data points. Comparing with Figure 5, we see that it gives a better fit than our previous linear model.

C (ppm)

340

350

360

370

380

390

400

1980 1985 t1990 1995 2000 2005 2010 ■

FIGURE 5� Linear model through first

and last data points

A computer or graphing calculator finds the regression line by the method of least squares, which is to minimize the sum of the squares of the vertical distances between the data points and the line. The details are explained in Section 14.7.

FIGURE 6� The regression line

Section 1.2 Mathematical Models: A Catalog of Essential Functions 27

ExamplE 3� Use the linear model given by Equa tion 2 to estimate the average CO2 level for 1987 and to predict the level for the year 2020. According to this model, when will the CO2 level exceed 420 parts per million?

SoLUtion Using Equation 2 with t − 1987, we estimate that the average CO2 level in 1987 was

Cs1987d − s1.71262ds1987d 2 3054.14 < 348.84

This is an example of interpolation because we have estimated a value between observed values. (In fact, the Mauna Loa Observatory reported that the average CO2 level in 1987 was 348.93 ppm, so our estimate is quite accurate.)

With t − 2020, we get

Cs2020d − s1.71262ds2020d 2 3054.14 < 405.35

So we predict that the average CO2 level in the year 2020 will be 405.4 ppm. This is an example of extrapolation because we have predicted a value outside the time frame of observations. Consequently, we are far less certain about the accuracy of our prediction.

Using Equation 2, we see that the CO2 level exceeds 420 ppm when

1.71262t 2 3054.14 . 420

Solving this inequality, we get

t .3474.14

1.71262< 2028.55

We therefore predict that the CO2 level will exceed 420 ppm by the year 2029. This pre diction is risky because it involves a time quite remote from our observations. In fact, we see from Figure 6 that the trend has been for CO2 levels to increase rather more rapidly in recent years, so the level might exceed 420 ppm well before 2029. ■

polynomialsA function P is called a polynomial if

Psxd − an xn 1 an21 xn21 1 ∙ ∙ ∙ 1 a2 x 2 1 a1 x 1 a0

where n is a nonnegative integer and the numbers a0, a1, a2, . . . , an are constants called the coefficients of the polynomial. The domain of any polynomial is R − s2`, `d. If the leading coefficient an ± 0, then the degree of the polynomial is n. For example, the function

Psxd − 2x 6 2 x 4 1 25 x 3 1 s2

is a polynomial of degree 6.A polynomial of degree 1 is of the form Psxd − mx 1 b and so it is a linear function.

A polynomial of degree 2 is of the form Psxd − ax 2 1 bx 1 c and is called a quadratic function. Its graph is always a parabola obtained by shifting the parabola y − ax 2, as we will see in the next section. The parabola opens upward if a . 0 and downward if a , 0. (See Figure 7.)

A polynomial of degree 3 is of the form

Psxd − ax 3 1 bx 2 1 cx 1 d a ± 0

FIGURE 7� The graphs of quadratic functions are parabolas.

y

2

x1

(a) y=≈+x+1

y

2

x1

(b) y=_2≈+3x+1

28 chapter 1 Functions and Models

and is called a cubic function. Figure 8 shows the graph of a cubic function in part (a) and graphs of polynomials of degrees 4 and 5 in parts (b) and (c). We will see later why the graphs have these shapes.

(a) y=˛-x+1

x

1

y

10

(b) y=x$-3≈+x

x

2

y

1

(c) y=3x%-25˛+60x

x

20

y

1

Polynomials are commonly used to model various quantities that occur in the natural and social sciences. For instance, in Section 3.7 we will explain why economists often use a polynomial Psxd to represent the cost of producing x units of a commodity. In the fol-lowing example we use a quadratic function to model the fall of a ball.

ExamplE 4� A ball is dropped from the upper observation deck of the CN Tower, 450 m above the ground, and its height h above the ground is recorded at 1-second intervals in Table 2. Find a model to fit the data and use the model to predict the time at which the ball hits the ground.

SoLUtion We draw a scatter plot of the data in Figure 9 and observe that a linear model is inappropriate. But it looks as if the data points might lie on a parabola, so we try a quadratic model instead. Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model:

3 h − 449.36 1 0.96t 2 4.90t 2

2

200

400

4 6 8 t0

200

400

t(seconds)

0 2 4 6 8

hh (meters)

In Figure 10 we plot the graph of Equation 3 together with the data points and see that the quadratic model gives a very good fit.

The ball hits the ground when h − 0, so we solve the quadratic equation

24.90t 2 1 0.96t 1 449.36 − 0

FIGURE 8�

Time (seconds)

Height (meters)

0 4501 4452 4313 4084 3755 3326 2797 2168 1439 61

Table 2

FIGURE 9 Scatter plot for a falling ball

FIGURE 10 Quadratic model for a falling ball

Section 1.2 Mathematical Models: A Catalog of Essential Functions 29

The quadratic formula gives

t −20.96 6 ss0.96d2 2 4s24.90d s449.36d

2s24.90d

The positive root is t < 9.67, so we predict that the ball will hit the ground after about 9.7 seconds. ■

power FunctionsA function of the form f sxd − xa, where a is a constant, is called a power function. We consider several cases.

(i ) a − n, where n is a positive integer

The graphs of f sxd − xn for n − 1, 2, 3, 4, and 5 are shown in Figure 11. (These are poly-nomials with only one term.) We already know the shape of the graphs of y − x (a line through the origin with slope 1) and y − x 2 [a parabola, see Example 1.1.2(b)].

x

1

y

10

y=x%

x

1

y

10

y=x#

x

1

y

10

y=≈

x

1

y

10

y=x

x

1

y

10

y=x$

The general shape of the graph of f sxd − xn depends on whether n is even or odd. If n is even, then f sxd − xn is an even function and its graph is similar to the parabola y − x 2. If n is odd, then f sxd − xn is an odd function and its graph is similar to that of y − x 3. Notice from Figure 12, however, that as n increases, the graph of y − xn becomes flatter near 0 and steeper when | x | > 1. (If x is small, then x 2 is smaller, x 3 is even smaller, x 4 is smaller still, and so on.)

y=x$

(1, 1)(_1, 1)

y=x^y=≈

(_1, _1)

(1, 1)

y

x

x

y

y=x#

y=x%

(i i) a − 1yn, where n is a positive integer

The function f sxd − x 1yn − sn x is a root function. For n − 2 it is the square root function f sxd − sx , whose domain is f0, `d and whose graph is the upper half of the

FIGURE 11 Graphs of f sxd − x n for n − 1, 2, 3, 4, 5

A family of functions is a collection of functions whose equations are related. Figure 12 shows two families of power functions, one with even powers and one with odd powers.

FIGURE 12

30 chapter 1 Functions and Models

parabola x − y 2. [See Figure 13(a).] For other even values of n, the graph of y − sn x is similar to that of y − sx . For n − 3 we have the cube root function f sxd − s3 x whose domain is R (recall that every real number has a cube root) and whose graph is shown in Figure 13(b). The graph of y − sn x for n odd sn . 3d is similar to that of y − s3 x .

(b) ƒ=Œ„x

x

y

(1, 1)

(a) ƒ=œ„x

x

y

(1, 1)

(iii) a − 21The graph of the reciprocal function f sxd − x21 − 1yx is shown in Figure 14. Its graph has the equation y − 1yx, or xy − 1, and is a hyperbola with the coordinate axes as its asymptotes. This function arises in physics and chemistry in connection with Boyle’s Law, which says that, when the temperature is constant, the volume V of a gas is inversely proportional to the pressure P:

V −C

P

where C is a constant. Thus the graph of V as a function of P (see Figure 15) has the same general shape as the right half of Figure 14.

Power functions are also used to model species-area relationships (Exercises 30–31), illumination as a function of distance from a light source (Exercise 29), and the period of revolution of a planet as a function of its distance from the sun (Exercise 32).

rational FunctionsA rational function f is a ratio of two polynomials:

f sxd −PsxdQsxd

where P and Q are polynomials. The domain consists of all values of x such that Qsxd ± 0. A simple example of a rational function is the function f sxd − 1yx, whose domain is hx | x ± 0j; this is the reciprocal function graphed in Figure 14. The function

f sxd −2x 4 2 x 2 1 1

x 2 2 4

is a rational function with domain hx | x ± 62j. Its graph is shown in Figure 16.

algebraic FunctionsA function f is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) start-ing with polynomials. Any rational function is automatically an algebraic function. Here are two more examples:

f sxd − sx 2 1 1 tsxd −x 4 2 16x 2

x 1 sx 1 sx 2 2ds3 x 1 1

FIGURE 13 Graphs of root functions

x

1

y

10

y=∆

FIGURE 14The reciprocal function

P

V

FIGURE 15�Volume as a function of pressure at constant temperature

ƒ=2x$-≈+1

≈-4

x

20

y

20

FIGURE 16�

Section 1.2 Mathematical Models: A Catalog of Essential Functions 31

When we sketch algebraic functions in Chapter 4, we will see that their graphs can assume a variety of shapes. Figure 17 illustrates some of the possibilities.

x

2

y

1

(a) ƒ=xœ„„„„x+3

x

1

y

50

(b) ©=$œ„„„„„„≈-25

x

1

y

10

(c) h(x)=x@?#(x-2)@

_3

An example of an algebraic function occurs in the theory of relativity. The mass of a particle with velocity v is

m − f svd −m0

s1 2 v 2yc 2

where m0 is the rest mass of the particle and c − 3.0 3 105 kmys is the speed of light in a vacuum.

trigonometric FunctionsTrigonometry and the trigonometric functions are reviewed on Reference Page 2 and also in Appendix D. In calculus the convention is that radian measure is always used (except when otherwise indicated). For example, when we use the function f sxd − sin x, it is understood that sin x means the sine of the angle whose radian measure is x. Thus the graphs of the sine and cosine functions are as shown in Figure 18.

(a) ƒ=sin x

π2

5π2

3π2

π2

_

x

y

π0_π

1

_12π 3π

(b) ©=cos x

x

y

1

_1

π_π

2π

3π

π2

5π2

3π2

π2

_

Notice that for both the sine and cosine functions the domain is s2`, `d and the range is the closed interval f21, 1g. Thus, for all values of x, we have

21 < sin x < 1 21 < cos x < 1

or, in terms of absolute values,

| sin x | < 1 | cos x | < 1

Also, the zeros of the sine function occur at the integer multiples of �; that is,

sin x − 0 when x − n� n an integer

FIGURE 17�

The Reference Pages are located at the back of the book.

FIGURE 18�

32 chapter 1 Functions and Models

An important property of the sine and cosine functions is that they are periodic func-tions and have period 2�. This means that, for all values of x,

sinsx 1 2�d − sin x cossx 1 2�d − cos x

The periodic nature of these functions makes them suitable for modeling repetitive phe-nomena such as tides, vibrating springs, and sound waves. For instance, in Example 1.3.4 we will see that a reasonable model for the number of hours of daylight in Philadelphia t days after January 1 is given by the function

Lstd − 12 1 2.8 sinF 2�

365st 2 80dG

ExamplE 5� What is the domain of the function f sxd −1

1 2 2 cos x?

SoLUtion This function is defined for all values of x except for those that make the denominator 0. But

1 2 2 cos x − 0 &? cos x −1

2 &? x −

�

3 1 2n� or x −

5�

3 1 2n�

where n is any integer (because the cosine function has period 2�). So the domain of f is the set of all real numbers except for the ones noted above. ■

The tangent function is related to the sine and cosine functions by the equation

tan x −sin x

cos x

and its graph is shown in Figure 19. It is undefined whenever cos x − 0, that is, when x − 6�y2, 63�y2, . . . . Its range is s2`, `d. Notice that the tangent function has per iod �:

tansx 1 �d − tan x for all x

The remaining three trigonometric functions (cosecant, secant, and cotangent) are the reciprocals of the sine, cosine, and tangent functions. Their graphs are shown in Appendix D.

exponential FunctionsThe exponential functions are the functions of the form f sxd − bx, where the base b is a positive constant. The graphs of y − 2x and y − s0.5dx are shown in Figure 20. In both cases the domain is s2`, `d and the range is s0, `d.

Exponential functions will be studied in detail in Section 1.4, and we will see that they are useful for modeling many natural phenomena, such as population growth (if b . 1) and radioactive decay (if b , 1d.

Logarithmic FunctionsThe logarithmic functions f sxd − logb x, where the base b is a positive constant, are the inverse functions of the exponential functions. They will be studied in Section 1.5. Figure

FIGURE 19y − tan xy=tan x

x

y

π0_π

1

π 2

3π 2

π 2

_3π 2

_

y

x

1

10

y

x1

10

(a) y=2® (b) y=(0.5)®

FIGURE 20

Section 1.2 Mathematical Models: A Catalog of Essential Functions 33

21 shows the graphs of four logarithmic functions with various bases. In each case the domain is s0, `d, the range is s2`, `d, and the function increases slowly when x . 1.

ExamplE 6� Classify the following functions as one of the types of functions that we have discussed.

(a) f sxd − 5x (b) tsxd − x 5

(c) hsxd −1 1 x

1 2 sx (d) ustd − 1 2 t 1 5t 4

SoLUtion

(a) f sxd − 5x is an exponential function. (The x is the exponent.)

(b) tsxd − x 5 is a power function. (The x is the base.) We could also consider it to be a polynomial of degree 5.

(c) hsxd −1 1 x

1 2 sx is an algebraic function.

(d) ustd − 1 2 t 1 5t 4 is a polynomial of degree 4. ■

1. 2 ExErcisEs

1–2 Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

1. (a) f sxd − log2 x (b) tsxd − s4 x

(c) hsxd −2x 3

1 2 x 2 (d) ustd − 1 2 1.1t 1 2.54t 2

(e) vstd − 5 t (f ) ws�d − sin � cos2�

2. (a) y − � x (b) y − x�

(c) y − x 2s2 2 x 3d (d) y − tan t 2 cos t

(e) y −s

1 1 s (f ) y −

sx 3 2 1

1 1 s3 x

3�–4� Match each equation with its graph. Explain your choices. (Don’t use a computer or graphing calculator.)

3�. (a) y − x 2 (b) y − x 5 (c) y − x 8

f

gh

y

x

4�. (a) y − 3x (b) y − 3x (c) y − x 3 (d) y − s3 x

G

f

g

Fy

x

5�–6� Find the domain of the function.

5�. f sxd −cos x

1 2 sin x 6�. tsxd −

1

1 2 tan x

7. (a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family.

(b) Find an equation for the family of linear functions such that f s2d − 1 and sketch several members of the family.

(c) Which function belongs to both families?

8. What do all members of the family of linear functions f sxd − 1 1 msx 1 3d have in common? Sketch several members of the family.

y

1

x1

y=log£ x

y=log™ x

y=log∞ xy=log¡¸ x

FIGURE 21

34 chapter 1 Functions and Models

9. What do all members of the family of linear functions f sxd − c 2 x have in common? Sketch several members of the family.

10. Find expressions for the quadratic functions whose graphs are shown.

y

(0, 1)

(1, _2.5)

(_2, 2)y

x0

(4, 2)

f

gx0

3

11. Find an expression for a cubic function f if f s1d − 6 and f s21d − f s0d − f s2d − 0.

12. Recent studies indicate that the average surface tempera- ture of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T − 0.02t 1 8.50, where T is temperature in °C and t represents years since 1900.

(a) What do the slope and T-intercept represent? (b) Use the equation to predict the average global surface

temperature in 2100.

13�. If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c − 0.0417Dsa 1 1d. Suppose the dosage for an adult is 200 mg.

(a) Find the slope of the graph of c. What does it represent? (b) What is the dosage for a newborn?

14�. The manager of a weekend flea market knows from past experience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equation y − 200 2 4x.

(a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can’t be negative quantities.)

(b) What do the slope, the y-intercept, and the x-intercept of the graph represent?

15�. The relationship between the Fahrenheit sFd and Celsius sCd temperature scales is given by the linear function F − 9

5 C 1 32. (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it repre-

sent? What is the F-intercept and what does it represent?

16�. Jason leaves Detroit at 2:00 pm and drives at a constant speed west along I-94. He passes Ann Arbor, 40 mi from Detroit, at 2:50 pm.

(a) Express the distance traveled in terms of the time elapsed.

(b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent?

17. Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relation-ship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70°F and 173 chirps per minute at 80°F.

(a) Find a linear equation that models the temperature T as a function of the number of chirps per minute N.

(b) What is the slope of the graph? What does it represent? (c) If the crickets are chirping at 150 chirps per minute,

estimate the temperature.

18. The manager of a furniture factory finds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day.

(a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph.

(b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it

represent?

19. At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lbyin2. Below the sur- face, the water pressure increases by 4.34 lbyin2 for every 10 ft of descent.

(a) Express the water pressure as a function of the depth below the ocean surface.

(b) At what depth is the pressure 100 lbyin2?

20. The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi.

(a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model.

(b) Use part (a) to predict the cost of driving 1500 miles per month.

(c) Draw the graph of the linear function. What does the slope represent?

(d) What does the C-intercept represent? (e) Why does a linear function give a suitable model in this

situation?

21–22 For each scatter plot, decide what type of function you might choose as a model for the data. Explain your choices.

21.

0 x

y(a)

0 x

y(b)

Section 1.2 Mathematical Models: A Catalog of Essential Functions 35

22.

0 x

y(a)

0 x

y(b)

23�. The table shows (lifetime) peptic ulcer rates (per 100 popula-tion) for various family incomes as reported by the National Health Interview Survey.

IncomeUlcer rate

(per 100 population)

$4,000 14.1$6,000 13.0$8,000 13.4

$12,000 12.5$16,000 12.0$20,000 12.4$30,000 10.5$45,000 9.4$60,000 8.2

(a) Make a scatter plot of these data and decide whether a linear model is appropriate.

(b) Find and graph a linear model using the first and last data points.

(c) Find and graph the least squares regression line. (d) Use the linear model in part (c) to estimate the ulcer

rate for an income of $25,000. (e) According to the model, how likely is someone with an

income of $80,000 to suffer from peptic ulcers? (f ) Do you think it would be reasonable to apply the model

to someone with an income of $200,000?

24�. Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures.

(a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the chirping

rate at 100°F.

Temperature (°F)

Chirping rate (chirpsymin)

Temperature (°F)

Chirping rate (chirpsymin)

50 20 75 14055 46 80 17360 79 85 19865 91 90 21170 113

;

;

25�. Anthropologists use a linear model that relates human femur (thighbone) length to height. The model allows an anthro-pologist to determine the height of an individual when only a partial skeleton (including the femur) is found. Here we find the model by analyzing the data on femur length and height for the eight males given in the following table.

(a) Make a scatter plot of the data. (b) Find and graph the regression line that models the data. (c) An anthropologist finds a human femur of length

53 cm. How tall was the person?

Femur length (cm)

Height (cm)

Femur length (cm)

Height (cm)

50.1 178.5 44.5 168.348.3 173.6 42.7 165.045.2 164.8 39.5 155.444.7 163.7 38.0 155.8

26�. When laboratory rats are exposed to asbestos fibers, some of them develop lung tumors. The table lists the results of several experiments by different scientists.

(a) Find the regression line for the data. (b) Make a scatter plot and graph the regression line.

Does the regression line appear to be a suitable model for the data?

(c) What does the y-intercept of the regression line represent?

Asbestos exposure

(fibersymL)

Percent of mice that develop lung tumors

Asbestos exposure

(fibersymL)

Percent of mice that develop lung tumors

50 2 1600 42400 6 1800 37500 5 2000 38900 10 3000 50

1100 26

27. The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day.

(a) Make a scatter plot and decide whether a linear model is appropriate.

(b) Find and graph the regression line. (c) Use the linear model to estimate the oil consumption in

2002 and 2012.

Years since 1985

Thousands of barrels of oil per day

0 60,0835 66,53310 70,09915 76,78420 84,07725 87,302

Source: US Energy Information Administration

;

;

;

36 chapter 1 Functions and Models

28. The table shows average US retail residential prices of electricity from 2000 to 2012, measured in cents per kilowatt hour.

(a) Make a scatter plot. Is a linear model appropriate? (b) Find and graph the regression line. (c) Use your linear model from part (b) to estimate the

average retail price of electricity in 2005 and 2013.

Years since 2000 CentsykWh

0 8.242 8.444 8.956 10.408 11.26

10 11.5412 11.58

Source: US Energy Information Administration

29. Many physical quantities are connected by inverse square laws, that is, by power functions of the form f sxd − kx22. In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light?

3�0. It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecolo-gists have modeled the species-area relation with a power function and, in particular, the number of species S of bats living in caves in central Mexico has been related to the surface area A of the caves by the equation S − 0.7A0.3.

(a) The cave called Misión Imposible near Puebla, Mexico, has a surface area of A − 60 m2. How many species of bats would you expect to find in that cave?

(b) If you discover that four species of bats live in a cave, estimate the area of the cave.

; 3�1. The table shows the number N of species of reptiles and amphibians inhabiting Caribbean islands and the area A of the island in square miles.

(a) Use a power function to model N as a function of A. (b) The Caribbean island of Dominica has area 291 mi2.

How many species of reptiles and amphibians would you expect to find on Dominica?

Island A N

Saba 4 5Monserrat 40 9Puerto Rico 3,459 40Jamaica 4,411 39Hispaniola 29,418 84Cuba 44,218 76

3�2. The table shows the mean (average) distances d of the planets from the sun (taking the unit of measurement to be the distance from the earth to the sun) and their periods T (time of revolution in years).

(a) Fit a power model to the data. (b) Kepler’s Third Law of Planetary Motion states that

“The square of the period of revolution of a planet is propor tional to the cube of its mean distance from the sun.” Does your model corroborate Kepler’s Third Law?

Planet d T

Mercury 0.387 0.241Venus 0.723 0.615Earth 1.000 1.000Mars 1.523 1.881Jupiter 5.203 11.861Saturn 9.541 29.457Uranus 19.190 84.008Neptune 30.086 164.784

;

;

In this section we start with the basic functions we discussed in Section 1.2 and obtain new functions by shifting, stretching, and reflecting their graphs. We also show how to combine pairs of functions by the standard arithmetic operations and by composition.

transformations of FunctionsBy applying certain transformations to the graph of a given function we can obtain the graphs of related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs.

Let’s first consider translations. If c is a positive number, then the graph of y − f sxd 1 c is just the graph of y − f sxd shifted upward a distance of c units (because each y-coordi-

Section 1.3 New Functions from Old Functions 37

nate is increased by the same number c). Likewise, if tsxd − f sx 2 cd, where c . 0, then the value of t at x is the same as the value of f at x 2 c (c units to the left of x). There- fore the graph of y − f sx 2 cd is just the graph of y − f sxd shifted c units to the right (see Figure 1).

Vertical and Horizontal Shifts Suppose c . 0. To obtain the graph of

y − f sxd 1 c, shift the graph of y − f sxd a distance c units upward

y − f sxd 2 c, shift the graph of y − f sxd a distance c units downward

y − f sx 2 cd, shift the graph of y − f sxd a distance c units to the right

y − f sx 1 cd, shift the graph of y − f sxd a distance c units to the left

y= ƒ1c

x

y

y=f(_x)

y=ƒ

y=_ƒ

y=cƒ(c>1)

x

y

y=f(x-c)y=f(x+c) y =ƒ

y=ƒ-c

y=ƒ+c

c

c

c c

Now let’s consider the stretching and reflecting transformations. If c . 1, then the graph of y − cf sxd is the graph of y − f sxd stretched by a factor of c in the vertical direction (because each y-coordinate is multiplied by the same number c). The graph of y − 2f sxd is the graph of y − f sxd reflected about the x-axis because the point sx, yd is replaced by the point sx, 2yd. (See Figure 2 and the following chart, where the results of other stretching, shrinking, and reflecting transformations are also given.)

Vertical and Horizontal Stretching and Reflecting Suppose c . 1. To obtain the graph of

y − cf sxd, stretch the graph of y − f sxd vertically by a factor of c

y − s1ycd f sxd, shrink the graph of y − f sxd vertically by a factor of c

y − f scxd, shrink the graph of y − f sxd horizontally by a factor of c

y − f sxycd, stretch the graph of y − f sxd horizontally by a factor of c

y − 2f sxd, reflect the graph of y − f sxd about the x-axis

y − f s2xd, reflect the graph of y − f sxd about the y-axis

FIGURE 2 Stretching and reflecting the graph of fFIGURE 1 Translating the graph of f

38 chapter 1 Functions and Models

Figure 3 illustrates these stretching transformations when applied to the cosine function with c − 2. For instance, in order to get the graph of y − 2 cos x we multiply the y-coordi-nate of each point on the graph of y − cos x by 2. This means that the graph of y − cos x gets stretched vertically by a factor of 2.

x

1

2

y

y=cos x

y=cos 2xy=cos x12

x

1

2

y

y=2 cos x

y=cos x

y= cos x12

1

ExamplE 1 Given the graph of y − sx , use transformations to graph y − sx 2 2,

y − sx 2 2 , y − 2sx , y − 2sx , and y − s2x .

SoLUtion The graph of the square root function y − sx , obtained from Fig- ure 1.2.13(a), is shown in Figure 4(a). In the other parts of the figure we sketch y − sx 2 2 by shifting 2 units downward, y − sx 2 2 by shifting 2 units to the right, y − 2sx by reflecting about the x-axis, y − 2sx by stretching vertically by a factor of 2, and y − s2x by reflecting about the y-axis.

(a) y=œ„x (b) y=œ„-2x (c) y=œ„„„„x-2 (d) y=_œ„x (e) y=2œ„x (f ) y=œ„„_x

0 x

y

0 x

y

0 x

y

20 x

y

_2

0 x

y

1

10 x

y

■

ExamplE 2 Sketch the graph of the function f sxd − x 2 1 6x 1 10.

SoLUtion Completing the square, we write the equation of the graph as

y − x 2 1 6x 1 10 − sx 1 3d2 1 1

This means we obtain the desired graph by starting with the parabola y − x 2 and shift-ing 3 units to the left and then 1 unit upward (see Figure 5).

(a) y=≈ (b) y=(x+3)@+1

x0_1_3

1

y

(_3, 1)

x0

y

■

FIGURE 3

FIGURE 4

FIGURE 5�

Section 1.3 New Functions from Old Functions 39

ExamplE 3� Sketch the graphs of the following functions.(a) y − sin 2x (b) y − 1 2 sin x

SoLUtion(a) We obtain the graph of y − sin 2x from that of y − sin x by compressing horizon-tally by a factor of 2. (See Figures 6 and 7.) Thus, whereas the period of y − sin x is 2�, the period of y − sin 2x is 2�y2 − �.

x0

y

1

π2

π4

π

y=sin 2x

FIGURE 7�

(b) To obtain the graph of y − 1 2 sin x, we again start with y − sin x. We reflect about the x-axis to get the graph of y − 2sin x and then we shift 1 unit upward to get y − 1 2 sin x. (See Figure 8.)

x

1

2

y

π0 2π

y=1-sin x

π2

3π2 ■

ExamplE 4� Figure 9 shows graphs of the number of hours of daylight as functions of the time of the year at several latitudes. Given that Philadelphia is located at approxi-mately 408N latitude, find a function that models the length of daylight at Philadelphia.

2

4

6

8

10

12

14

16

18

20

Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.

Hours

60° N

50° N40° N30° N20° N

FIGURE 6�

x0

y

1

π2

π

y=sin x

FIGURE 8�

FIGURE 9 Graph of the length of daylight from

March 21 through December 21 at various latitudes

Source: Adapted from L. Harrison, Daylight, Twilight, Darkness and Time (New York: Silver, Burdett, 1935), 40.

40 chapter 1 Functions and Models

SoLUtion Notice that each curve resembles a shifted and stretched sine function. By looking at the blue curve we see that, at the latitude of Philadelphia, daylight lasts about 14.8 hours on June 21 and 9.2 hours on December 21, so the amplitude of the curve (the factor by which we have to stretch the sine curve vertically) is 12 s14.8 2 9.2d − 2.8.

By what factor do we need to stretch the sine curve horizontally if we measure the time t in days? Because there are about 365 days in a year, the period of our model should be 365. But the period of y − sin t is 2�, so the horizontal stretching factor is 2�y365.

We also notice that the curve begins its cycle on March 21, the 80th day of the year, so we have to shift the curve 80 units to the right. In addition, we shift it 12 units upward. Therefore we model the length of daylight in Philadelphia on the tth day of the year by the function

Lstd − 12 1 2.8 sinF 2�

365st 2 80dG

■

Another transformation of some interest is taking the absolute value of a function. If y − | f sxd|, then according to the definition of absolute value, y − f sxd when f sxd > 0 and y − 2f sxd when f sxd , 0. This tells us how to get the graph of y − | f sxd| from the graph of y − f sxd: The part of the graph that lies above the x-axis remains the same; the part that lies below the x-axis is reflected about the x-axis.

ExamplE 5� Sketch the graph of the function y − | x 2 2 1 |.SoLUtion We first graph the parabola y − x 2 2 1 in Figure 10(a) by shifting the parabola y − x 2 downward 1 unit. We see that the graph lies below the x-axis when 21 , x , 1, so we reflect that part of the graph about the x-axis to obtain the graph of y − | x 2 2 1| in Figure 10(b). ■

combinations of FunctionsTwo functions f and t can be combined to form new functions f 1 t, f 2 t, ft, and fyt in a manner similar to the way we add, subtract, multiply, and divide real numbers. The sum and difference functions are defined by

s f 1 tdsxd − f sxd 1 tsxd s f 2 tdsxd − f sxd 2 tsxd

If the domain of f is A and the domain of t is B, then the domain of f 1 t is the inter-section A > B because both f sxd and tsxd have to be defined. For example, the domainof f sxd − sx is A − f0, `d and the domain of tsxd − s2 2 x is B − s2`, 2g, so thedomain of s f 1 tdsxd − sx 1 s2 2 x is A > B − f0, 2g.

Similarly, the product and quotient functions are defined by

s ftdsxd − f sxdtsxd S f

tDsxd − f sxdtsxd

The domain of ft is A > B, but we can’t divide by 0 and so the domain of fyt is hx [ A > B | tsxd ± 0j. For instance, if f sxd − x 2 and tsxd − x 2 1, then the domain of the rational function s fytdsxd − x 2ysx 2 1d is hx | x ± 1j, or s2`, 1d ø s1, `d.

There is another way of combining two functions to obtain a new function. Forexample, suppose that y − f sud − su and u − tsxd − x 2 1 1. Since y is a function of u and u is, in turn, a function of x, it follows that y is ultimately a function of x.

FIGURE 10

0 x

y

_1 1

(a) y=≈-1

(b) y=| ≈-1 |

0 x

y

_1 1

Section 1.3 New Functions from Old Functions 41

We compute this by substitution:

y − f sud − f stsxdd − f sx 2 1 1d − sx 2 1 1

The procedure is called composition because the new function is composed of the two given functions f and t.

In general, given any two functions f and t, we start with a number x in the domain of t and calculate tsxd. If this number tsxd is in the domain of f, then we can calculate the value of f stsxdd. Notice that the output of one function is used as the input to the next function. The result is a new function hsxd − f stsxdd obtained by substituting t into f. It is called the composition (or composite) of f and t and is denoted by f 8 t (“ f circle t”).

Definition Given two functions f and t, the composite function f 8 t (also called the composition of f and t) is defined by

s f 8 tdsxd − f stsxdd

The domain of f 8 t is the set of all x in the domain of t such that tsxd is in the domain of f. In other words, s f 8 tdsxd is defined whenever both tsxd and f stsxdd are defined. Figure 11 shows how to picture f 8 t in terms of machines.

ExamplE 6� If f sxd − x 2 and tsxd − x 2 3, find the composite functions f 8 t and t 8 f.

SoLUtion We have

s f 8 tdsxd − f stsxdd − f sx 2 3d − sx 2 3d2

st 8 f dsxd − ts f sxdd − tsx 2 d − x 2 2 3 ■

NOTE You can see from Example 6 that, in general, f 8 t ± t 8 f. Remember, the notation f 8 t means that the function t is applied first and then f is applied second. In Example 6, f 8 t is the function that first subtracts 3 and then squares; t 8 f is the function that first squares and then subtracts 3.

ExamplE 7 If f sxd − sx and tsxd − s2 2 x , find each of the following functions and their domains.(a) f 8 t (b) t 8 f (c) f 8 f (d) t 8 t

SoLUtion(a) s f 8 tdsxd − f stsxdd − f (s2 2 x) − ss2 2 x − s4 2 2 x

The domain of f 8 t is hx | 2 2 x > 0j − hx | x < 2j − s2`, 2g.

(b) st 8 f dsxd − ts f sxdd − t(sx ) − s2 2 sx

For sx to be defined we must have x > 0. For s2 2 sx to be defined we must have2 2 sx > 0, that is, sx < 2, or x < 4. Thus we have 0 < x < 4, so the domain of t 8 f is the closed interval f0, 4g.

(c) s f 8 f dsxd − f s f sxdd − f (sx ) − ssx − s4 x

The domain of f 8 f is f0, `d.

FIGURE 11 The f 8 t machine is composed of the t machine (first) and then the f machine.

f

g

f{©}

f • g

x

©

(input)

(output)

If 0 < a < b, then a 2 < b 2.

42 chapter 1 Functions and Models

(a) y − f sx 2 4d (b) y − f sxd 1 3 (c) y − 1

3 f sxd (d) y − 2f sx 1 4d (e) y − 2 f sx 1 6d

!@

$

%

#f

y

3

_3

6

0 x3_3_6 6

1. Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows.

(a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Reflect about the x-axis. (f ) Reflect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3.

2. Explain how each graph is obtained from the graph of y − f sxd. (a) y − f sxd 1 8 (b) y − f sx 1 8d (c) y − 8 f sxd (d) y − f s8xd (e) y − 2f sxd 2 1 (f) y − 8 f s 1

8 xd 3�. The graph of y − f sxd is given. Match each equation with its

graph and give reasons for your choices.

(d) st 8 tdsxd − tstsxdd − t(s2 2 x ) − s2 2 s2 2 x

This expression is defined when both 2 2 x > 0 and 2 2 s2 2 x > 0. The first inequality means x < 2, and the second is equivalent to s2 2 x < 2, or 2 2 x < 4, or x > 22. Thus 22 < x < 2, so the domain of t 8 t is the closed interval f22, 2g. ■

It is possible to take the composition of three or more functions. For instance, the composite function f 8 t 8 h is found by first applying h, then t, and then f as follows:

s f 8 t 8 hdsxd − f stshsxddd

ExamplE 8 Find f 8 t 8 h if f sxd − xysx 1 1d, tsxd − x 10, and hsxd − x 1 3.

SoLUtion s f 8 t 8 hdsxd − f stshsxddd − f stsx 1 3dd

− f ssx 1 3d10 d −sx 1 3d10

sx 1 3d10 1 1 ■

So far we have used composition to build complicated functions from simpler ones. But in calculus it is often useful to be able to decompose a complicated function into simpler ones, as in the following example.

ExamplE 9 Given Fsxd − cos2sx 1 9d, find functions f , t, and h such that F − f 8 t 8 h.

SoLUtion Since Fsxd − fcossx 1 9dg2, the formula for F says: First add 9, then take the cosine of the result, and finally square. So we let

hsxd − x 1 9 tsxd − cos x f sxd − x 2

Then s f 8 t 8 hdsxd − f stshsxddd − f stsx 1 9dd − f scossx 1 9dd

− fcossx 1 9dg2 − Fsxd ■

1. 3 ExErcisEs

Section 1.3 New Functions from Old Functions 43

4�. The graph of f is given. Draw the graphs of the following functions.

(a) y − f sxd 2 3 (b) y − f sx 1 1d

(c) y − 12 f sxd (d) y − 2f sxd

x

y

0 1

2

5�. The graph of f is given. Use it to graph the following functions.

(a) y − f s2xd (b) y − f s 12xd

(c) y − f s2xd (d) y − 2f s2xd

x

y

0 1

1

6�–7 The graph of y − s3x 2 x 2 is given. Use transformations to create a function whose graph is as shown.

1.5 y=œ„„„„„„3x-≈

x

y

30

6�.

5 x

y

20

3

7.

_4_1

_2.5

x

y

_1 0

8. (a) How is the graph of y − 2 sin x related to the graph of y − sin x? Use your answer and Figure 6 to sketch the graph of y − 2 sin x.

(b) How is the graph of y − 1 1 sx related to the graph of y − sx

? Use your answer and Figure 4(a) to sketch the graph of y − 1 1 sx .

9–24� Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

9. y − 2x 2 10. y − sx 2 3d2

11. y − x 3 1 1 12. y − 1 21

x

13�. y − 2 cos 3x 14�. y − 2sx 1 1

15�. y − x 2 2 4x 1 5 16�. y − 1 1 sin �x

17. y − 2 2 sx 18. y − 3 2 2 cos x

19. y − sin( 12 x) 20. y − | x | 2 2

21. y − | x 2 2 | 22. y −1

4 tanSx 2

�

4 D 23�. y − | sx 2 1 | 24�. y − | cos �x |

25�. The city of New Orleans is located at latitude 30°N. Use Figure 9 to find a function that models the number of hours of daylight at New Orleans as a function of the time of year. To check the accuracy of your model, use the fact that on March 31 the sun rises at 5:51 am and sets at 6:18 pm in New Orleans.

26�. A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by 60.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time.

27. Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on June 30, 2009, high tide occurred at 6:45 am. Find a function involving the cosine function that models the water depth Dstd (in meters) as a function of time t (in hours after midnight) on that day.

28. In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residue volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 seconds. Find a model for the total volume of air Vstd in the lungs as a function of time.

29. (a) How is the graph of y − f (| x |) related to the graph of f ? (b) Sketch the graph of y − sin | x |. (c) Sketch the graph of y − s| x |.

44 Chapter 1 Functions and Models

52. Use the table to evaluate each expression. (a) f sts1dd (b) ts f s1dd (c) f s f s1dd (d) tsts1dd (e) st 8 f ds3d (f ) s f 8 tds6d

x 1 2 3 4 5 6

f sxd 3 1 4 2 2 5

tsxd 6 3 2 1 2 3

53. Use the given graphs of f and t to evaluate each expression, or explain why it is undefined.

(a) f sts2dd (b) ts f s0dd (c) s f 8 tds0d (d) st 8 f ds6d (e) st 8 tds22d (f ) s f 8 f ds4d

x

y

fg

2

2

54. Use the given graphs of f and t to estimate the value of f stsxdd for x − 25, 24, 23, . . . , 5. Use these estimates to sketch a rough graph of f 8 t.

g

f

x

y

0 1

1

55. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cmys.

(a) Express the radius r of this circle as a function of the time t (in seconds).

(b) If A is the area of this circle as a function of the radius, find A 8 r and interpret it.

56. A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cmys.

(a) Express the radius r of the balloon as a function of the time t (in seconds).

(b) If V is the volume of the balloon as a function of the radius, find V 8 r and interpret it.

57. A ship is moving at a speed of 30 kmyh parallel to a straight shoreline. The ship is 6 km from shore and it passes a light-house at noon.

(a) Express the distance s between the lighthouse and the ship

30. Use the given graph of f to sketch the graph of y − 1yf sxd. Which features of f are the most important in sketching y − 1yf sxd? Explain how they are used.

1

10 x

y

31–32 Find (a) f 1 t, (b) f 2 t, (c) ft, and (d) fyt and state their domains.

31. f sxd − x 3 1 2x 2, tsxd − 3x 2 2 1

32. f sxd − s3 2 x , tsxd − sx 2 2 1

33–38 Find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , and (d) t 8 t and their domains.

33. f sxd − 3x 1 5, tsxd − x 2 1 x

34. f sxd − x 3 2 2, tsxd − 1 2 4x

35. f sxd − sx 1 1, tsxd − 4x 2 3

36. f sxd − sin x, tsxd − x 2 1 1

37. f sxd − x 11

x, tsxd −

x 1 1

x 1 2

38. f sxd −x

1 1 x, tsxd − sin 2x

39–42 Find f 8 t 8 h.

39. f sxd − 3x 2 2, tsxd − sin x, hsxd − x 2

40. f sxd − | x 2 4 |, tsxd − 2 x, hsxd − sx

41. f sxd − sx 2 3 , tsxd − x 2, hsxd − x 3 1 2

42. f sxd − tan x, tsxd −x

x 2 1, hsxd − s3 x

43–48 Express the function in the form f 8 t.

43. Fsxd − s2x 1 x 2d4 44. Fsxd − cos2x

45. Fsxd −s3 x

1 1 s3 x 46. Gsxd − 3Î x

1 1 x

47. vstd − secst 2d tanst 2d 48. ustd −tan t

1 1 tan t

49–51 Express the function in the form f 8 t 8 h.

49. Rsxd − ssx 2 1 50. Hsxd − s8 2 1 | x | 51. Sstd − sin2scos td

Section 1.4 Exponential Functions 45

In Appendix G we present an alterna-tive approach to the exponential and logarithmic functions using integral calculus.

as a function of d, the distance the ship has traveled since noon; that is, find f so that s − f sdd.

(b) Express d as a function of t, the time elapsed since noon; that is, find t so that d − tstd.

(c) Find f 8 t. What does this function represent?

5�8. An airplane is flying at a speed of 350 miyh at an altitude of one mile and passes directly over a radar station at time t − 0.

(a) Express the horizontal distance d (in miles) that the plane has flown as a function of t.

(b) Express the distance s between the plane and the radar station as a function of d.

(c) Use composition to express s as a function of t.

5�9. The Heaviside function H is defined by

Hstd − H0

1

if t , 0

if t > 0

It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on.

(a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage Vstd in a circuit if the

switch is turned on at time t − 0 and 120 volts are applied instantaneously to the circuit. Write a formula for Vstd in terms of Hstd.

(c) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for Vstd in terms of Hstd. (Note that starting at t − 5 corre sponds to a translation.)

6�0. The Heaviside function defined in Exercise 59 can also be used to define the ramp function y − ctHstd, which

The function f sxd − 2x is called an exponential function because the variable, x, is the exponent. It should not be confused with the power function tsxd − x 2, in which the variable is the base.

In general, an exponential function is a function of the form

f sxd − bx

where b is a positive constant. Let’s recall what this means.If x − n, a positive integer, then

bn − b ? b ? ∙ ∙ ∙ ? b

n factors

If x − 0, then b 0 − 1, and if x − 2n, where n is a positive integer, then

b2n −1

bn

represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y − tHstd. (b) Sketch the graph of the voltage Vstd in a circuit if the

switch is turned on at time t − 0 and the voltage is gradu-ally increased to 120 volts over a 60-second time interval. Write a formula for Vstd in terms of Hstd for t < 60.

(c) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t − 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 sec-onds. Write a formula for Vstd in terms of Hstd for t < 32.

6�1. Let f and t be linear functions with equations f sxd − m1x 1 b1 and tsxd − m2 x 1 b2. Is f 8 t also a linear function? If so, what is the slope of its graph?

6�2. If you invest x dollars at 4% interest compounded annually, then the amount Asxd of the investment after one year is Asxd − 1.04x. Find A 8 A, A 8 A 8 A, and A 8 A 8 A 8 A. What do these compositions represent? Find a formula for the com-position of n copies of A.

6�3�. (a) If tsxd − 2x 1 1 and hsxd − 4x 2 1 4x 1 7, find a func-tion f such that f 8 t − h. (Think about what operations you would have to perform on the formula for t to end up with the formula for h.)

(b) If f sxd − 3x 1 5 and hsxd − 3x 2 1 3x 1 2, find a func-tion t such that f 8 t − h.

6�4�. If f sxd − x 1 4 and hsxd − 4x 2 1, find a function t such that t 8 f − h.

6�5�. Suppose t is an even function and let h − f 8 t. Is h always an even function?

6�6�. Suppose t is an odd function and let h − f 8 t. Is h always an odd function? What if f is odd? What if f is even?

46 chapter 1 Functions and Models

If x is a rational number, x − pyq, where p and q are integers and q . 0, then

bx − bpyq − sq bp − ssq b d p

But what is the meaning of bx if x is an irrational number? For instance, what is meant by 2s3 or 5�?

To help us answer this question we first look at the graph of the function y − 2x, where x is rational. A representation of this graph is shown in Figure 1. We want to enlarge the domain of y − 2x to include both rational and irrational numbers.

There are holes in the graph in Figure 1 corresponding to irrational values of x. We want to fill in the holes by defining f sxd − 2x, where x [ R, so that f is an increasing function. In particular, since the irrational number s3 satisfies

1.7 , s3 , 1.8

we must have 21.7 , 2s3 , 21.8

and we know what 21.7 and 21.8 mean because 1.7 and 1.8 are rational numbers. Similarly, if we use better approximations for s3 , we obtain better approximations for 2s3:

1.73 , s3 , 1.74 ? 21.73 , 2s3 , 21.74

1.732 , s3 , 1.733 ? 21.732 , 2s3 , 21.733

1.7320 , s3 , 1.7321 ? 21.7320 , 2s3 , 21.7321

1.73205 , s3 , 1.73206 ? 21.73205 , 2s3 , 21.73206

. . . . . . . . . . . .

It can be shown that there is exactly one number that is greater than all of the numbers

21.7, 21.73, 21.732, 21.7320, 21.73205, . . .

and less than all of the numbers

21.8, 21.74, 21.733, 21.7321, 21.73206, . . .

We define 2s3 to be this number. Using the preceding approximation process we can compute it correct to six decimal places:

2s3 < 3.321997

Similarly, we can define 2x (or bx, if b . 0) where x is any irrational number. Figure 2 shows how all the holes in Figure 1 have been filled to complete the graph of the function f sxd − 2x, x [ R.

x10

y

1

A proof of this fact is given in J. Marsden and A. Weinstein, Calculus Unlimited (Menlo Park, CA: Benjamin/Cummings, 1981).

FIGURE 2 y − 2 x, x real

FIGURE 1 Representation of y − 2x, x rational

x0

y

1

1

Section 1.4 Exponential Functions 47

The graphs of members of the family of functions y − bx are shown in Figure 3 for various values of the base b. Notice that all of these graphs pass through the same point s0, 1d because b 0 − 1 for b ± 0. Notice also that as the base b gets larger, the exponential function grows more rapidly (for x . 0).

1®

1.5®2®4®10®” ’®14” ’®1

2

x

y

1

You can see from Figure 3 that there are basically three kinds of exponential functions y − bx. If 0 , b , 1, the exponential function decreases; if b − 1, it is a constant; and if b . 1, it increases. These three cases are illustrated in Figure 4. Observe that if b ± 1, then the exponential function y − bx has domain R and range s0, `d. Notice also that, since s1ybdx − 1ybx − b2x, the graph of y − s1ybdx is just the reflection of the graph of y − bx about the y-axis.

(a) y=b®, 0<b<1 (b) y=1® (c) y=b®, b>1

1(0, 1)

(0, 1)

x0

y y

x0x0

y

One reason for the importance of the exponential function lies in the following proper-ties. If x and y are rational numbers, then these laws are well known from elementary algebra. It can be proved that they remain true for arbitrary real numbers x and y.

Laws of Exponents If a and b are positive numbers and x and y are any real numbers, then

1. bx1y − bxby 2. bx2y −bx

by 3. sbx dy − bxy 4. sabdx − axbx

ExamplE 1 Sketch the graph of the function y − 3 2 2x and determine its domain and range.

SoLUtion First we reflect the graph of y − 2x [shown in Figures 2 and 5(a)] about the x-axis to get the graph of y − 22x in Figure 5(b). Then we shift the graph of y − 22x

FIGURE 3

If 0 , b , 1, then b x approaches 0 as x becomes large. If b . 1, then b x approaches 0 as x decreases through negative values. In both cases the x-axis is a horizontal asymptote. These matters are discussed in Sec tion 2.6.

FIGURE 4

www.stewartcalculus.comFor review and practice using the Laws of Exponents, click on Review of Algebra.

For a review of reflecting and shifting graphs, see Section 1.3.

48 chapter 1 Functions and Models

upward 3 units to obtain the graph of y − 3 2 2x in Figure 5(c). The domain is R and the range is s2`, 3d.

1

(a) y=2®

x

y

_1

(b) y=_2®

x

y

y=3

2

(c) y=3-2®

x

y

■

ExamplE 2 Use a graphing device to compare the exponential function f sxd − 2x and the power function tsxd − x 2. Which function grows more quickly when x is large?

SoLUtion Figure 6 shows both functions graphed in the viewing rectangle f22, 6g by f0, 40g. We see that the graphs intersect three times, but for x . 4 the graph of f sxd − 2x stays above the graph of tsxd − x 2. Figure 7 gives a more global view and shows that for large values of x, the exponential function y − 2x grows far more rapidly than the power function y − x 2.

250

0 8

y=2®

y=≈

40

0_2 6

y=2® y=≈

FIGURE 6� FIGURE 7� ■

applications of exponential FunctionsThe exponential function occurs very frequently in mathematical models of nature and society. Here we indicate briefly how it arises in the description of population growth and radioactive decay. In later chapters we will pursue these and other applications in greater detail.

First we consider a population of bacteria in a hom*ogeneous nutrient medium. Suppose that by sampling the population at certain intervals it is determined that the population doubles every hour. If the number of bacteria at time t is pstd, where t is measured in hours, and the initial population is ps0d − 1000, then we have

ps1d − 2ps0d − 2 3 1000

ps2d − 2ps1d − 22 3 1000

ps3d − 2ps2d − 23 3 1000

FIGURE 5�

Example 2 shows that y − 2x increases more quickly than y − x 2. To demon-strate just how quickly f sxd − 2x increases, let’s perform the following thought experiment. Suppose we start with a piece of paper a thousandth of an inch thick and we fold it in half 50 times. Each time we fold the paper in half, the thickness of the paper doubles, so the thickness of the resulting paper would be 250y1000 inches. How thick do you think that is? It works out to be more than 17 million miles!

Section 1.4 Exponential Functions 49

It seems from this pattern that, in general,

pstd − 2t 3 1000 − s1000d2 t

This population function is a constant multiple of the exponential function y − 2t, so it exhibits the rapid growth that we observed in Figures 2 and 7. Under ideal conditions (unlimited space and nutrition and absence of disease) this exponential growth is typical of what actually occurs in nature.

What about the human population? Table 1 shows data for the population of the world in the 20th century and Figure 8 shows the corresponding scatter plot.

5x10'

P

t20 40 60 80 100 1200

Years since 1900

The pattern of the data points in Figure 8 suggests exponential growth, so we use a graph- ing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model

P − s1436.53d ? s1.01395d t

where t − 0 corresponds to 1900. Figure 9 shows the graph of this exponential function together with the original data points. We see that the exponential curve fits the data rea-sonably well. The period of relatively slow population growth is explained by the two world wars and the Great Depression of the 1930s.

5x10'

20 40 60 80 100 120

P

t0

Years since 1900

figure 8 Scatter plot for world

population growth

figure 9 Exponential model for

population growth

t(years since 1900)

Population (millions)

0 165010 175020 186030 207040 230050 256060 304070 371080 445090 5280

100 6080110 6870

Table 1�

50 chapter 1 Functions and Models

In 1995 a paper appeared detailing the effect of the protease inhibitor ABT-538 on the human immunodeficiency virus HIV-1.1 Table 2 shows values of the plasma viral load Vstd of patient 303, measured in RNA copies per mL, t days after ABT-538 treatment was begun. The corresponding scatter plot is shown in Figure 10.

RN

A c

opie

s / m

L

V

10 20 30

20

40

60

t (days)

FIGURE 10 Plasma viral load in patient 303

The rather dramatic decline of the viral load that we see in Figure 10 reminds us of the graphs of the exponential function y − bx in Figures 3 and 4(a) for the case where the base b is less than 1. So let’s model the function Vstd by an exponential function. Using a graphing calculator or computer to fit the data in Table 2 with an exponential function of the form y − a ? bt, we obtain the model

V − 96.39785 ? s0.818656dt

In Figure 11 we graph this exponential function with the data points and see that the model represents the viral load reasonably well for the first month of treatment.

RN

A c

opie

s / m

L

V

10 20 30

20

40

60

t (days)

We could use the graph in Figure 11 to estimate the half-life of V, that is, the time required for the viral load to be reduced to half its initial value (see Exercise 33). In the next example we are given the half-life of a radioactive element and asked to find the mass of a sample at any time.

ExamplE 3� The half-life of strontium-90, 90Sr, is 25 years. This means that half of any given quantity of 90Sr will disintegrate in 25 years.(a) If a sample of 90Sr has a mass of 24 mg, find an expression for the mass mstd that remains after t years.(b) Find the mass remaining after 40 years, correct to the nearest milligram.(c) Use a graphing device to graph mstd and use the graph to estimate the time required for the mass to be reduced to 5 mg.

Table 2

t (days) Vstd

1 76.0

4 53.0

8 18.0

11 9.4

15 5.2

22 3.6

FIGURE 11 Exponential model for viral load

1. D. Ho et al., “Rapid Turnover of Plasma Virions and CD4 Lymphocytes in HIV-1 Infection,” Nature 373 (1995): 123–26.

Section 1.4 Exponential Functions 51

SoLUtion (a) The mass is initially 24 mg and is halved during each 25-year period, so

ms0d − 24

ms25d −1

2s24d

ms50d −1

2?

1

2s24d −

1

22 s24d

ms75d −1

2?

1

22 s24d −1

23 s24d

ms100d −1

2?

1

23 s24d −1

24 s24d

From this pattern, it appears that the mass remaining after t years is

mstd −1

2ty25 s24d − 24 ? 22ty25 − 24 ? s221y25dt

This is an exponential function with base b − 221y25 − 1y21y25.

(b) The mass that remains after 40 years is

ms40d − 24 ? 2240y25 < 7.9 mg

(c) We use a graphing calculator or computer to graph the function mstd − 24 ? 22ty25 in Figure 12. We also graph the line m − 5 and use the cursor to estimate that mstd − 5 when t < 57. So the mass of the sample will be reduced to 5 mg after about 57 years. ■

the number eOf all possible bases for an exponential function, there is one that is most convenient for the purposes of calculus. The choice of a base b is influenced by the way the graph of y − bx crosses the y-axis. Figures 13 and 14 show the tangent lines to the graphs of y − 2x and y − 3x at the point s0, 1d. (Tangent lines will be defined precisely in Section 2.7. For present purposes, you can think of the tangent line to an exponential graph at a point as the line that touches the graph only at that point.) If we measure the slopes of these tangent lines at s0, 1d, we find that m < 0.7 for y − 2x and m < 1.1 for y − 3x.

1

mÅ1.1

y=2®

1mÅ0.7

x

yy=3®

x

y

FIGURE 13 FIGURE 14

It turns out, as we will see in Chapter 3, that some of the formulas of calculus will be greatly simplified if we choose the base b so that the slope of the tangent line to y − bx

m=24 · 2_t/25

m=5

30

0 100

FIGURE 12

52 Chapter 1 Functions and Models

at s0, 1d is exactly 1. (See Figure 15.) In fact, there is such a number and it is denoted by the letter e. (This notation was chosen by the Swiss mathematician Leonhard Euler in 1727, probably because it is the first letter of the word exponential.) In view of Figures 13 and 14, it comes as no surprise that the number e lies between 2 and 3 and the graph of y − ex lies between the graphs of y − 2x and y − 3x. (See Figure 16.) In Chapter 3 we will see that the value of e, correct to five decimal places, is

e < 2.71828

We call the function f sxd − ex the natural exponential function.

1

y=2®

y=e®

y=3®y

x

ExamplE 4� Graph the function y − 12 e2x 2 1 and state the domain and range.

SOLUtION We start with the graph of y − ex from Figures 15 and 17(a) and reflect about the y-axis to get the graph of y − e2x in Figure 17(b). (Notice that the graph crosses the y-axis with a slope of 21). Then we compress the graph vertically by a factor of 2 to obtain the graph of y − 1

2 e2x in Figure 17(c). Finally, we shift the graph downward one unit to get the desired graph in Figure 17(d). The domain is R and the range is s21, `d.

12(d) y= e–®-1

y=_10

1

12(c) y= e–®

1

(b) y=e–®

1

x0

y

(a) y=´

1

y

x

y

x

y

x

■

How far to the right do you think we would have to go for the height of the graph of y − ex to exceed a million? The next example demonstrates the rapid growth of this function by providing an answer that might surprise you.

ExamplE 5� Use a graphing device to find the values of x for which ex . 1,000,000.

y=´

1

m=1

x

y

FIGURE 15 The natural exponential function crosses the y-axis with a slope of 1.

FIGURE 16

FIGURE 17

TEC Module 1.4 enables you to graph exponential functions with various bases and their tangent lines in order to estimate more closely the value of b for which the tangent has slope 1.

Section 1.4 Exponential Functions 53

SoLUtion In Figure 18 we graph both the function y − ex and the horizontal line y − 1,000,000. We see that these curves intersect when x < 13.8. Thus ex . 106 when x . 13.8. It is perhaps surprising that the values of the exponential function have already surpassed a million when x is only 14.

1.5x10^

0 15

y=´

y=10^

■FIGURE 18

1–4 Use the Law of Exponents to rewrite and simplify the expression.

1. (a) 423

228 (b) 1

s3 x 4

2. (a) 8 4y3 (b) xs3x 2d3

3. (a) b8s2bd4 (b) s6y3d4

2y 5

4. (a) x 2n ? x 3n21

x n12 (b) sasb

s3 ab

5. (a) Write an equation that defines the exponential function with base b . 0.

(b) What is the domain of this function? (c) If b ± 1, what is the range of this function? (d) Sketch the general shape of the graph of the exponential

function for each of the following cases. (i) b . 1 (ii) b − 1 (iii) 0 , b , 1

6. (a) How is the number e defined? (b) What is an approximate value for e? (c) What is the natural exponential function?

7–10 Graph the given functions on a common screen. How are these graphs related?

7. y − 2x, y − e x, y − 5x, y − 20 x

8. y − e x, y − e 2x, y − 8x, y − 82x

;

9. y − 3x, y − 10 x, y − ( 13)x

, y − ( 110 )x

10. y − 0.9 x, y − 0.6x, y − 0.3x, y − 0.1x

11–16 Make a rough sketch of the graph of the function. Do not use a calculator. Just use the graphs given in Figures 3 and 13 and, if necessary, the transformations of Section 1.3.

11. y − 4x 2 1 12. y − s0.5dx 21

13. y − 222x 14. y − e | x |

15. y − 1 2 12 e2x 16. y − 2s1 2 e x d

17. Starting with the graph of y − e x, write the equation of the graph that results from

(a) shifting 2 units downward. (b) shifting 2 units to the right. (c) reflecting about the x-axis. (d) reflecting about the y-axis. (e) reflecting about the x-axis and then about the y-axis.

18. Starting with the graph of y − e x, find the equation of the graph that results from

(a) reflecting about the line y − 4. (b) reflecting about the line x − 2.

19–20 Find the domain of each function.

19. (a) f sxd −1 2 e x 2

1 2 e12x 2 (b) f sxd −1 1 x

e cos x

20. (a) tstd − s10 t 2 100 (b) tstd − sinse t 2 1d

54 Chapter 1 Functions and Models

21–22 Find the exponential function f sxd − Cb x whose graph is given.

21. 22.

(1, 6)

(3, 24)y

x

(_1, 3)

”1, ’43

y

x

23. If f sxd − 5x, show that

f sx 1 hd 2 f sxdh

− 5xS 5h 2 1

h D 24�. Suppose you are offered a job that lasts one month. Which

of the following methods of payment do you prefer? I. One million dollars at the end of the month. II. One cent on the first day of the month, two cents on the

second day, four cents on the third day, and, in general, 2n21 cents on the nth day.

25�. Suppose the graphs of f sxd − x 2 and tsxd − 2x are drawn on a coordinate grid where the unit of measurement is 1 inch. Show that, at a distance 2 ft to the right of the origin, the height of the graph of f is 48 ft but the height of the graph of t is about 265 mi.

26. Compare the functions f sxd − x 5 and tsxd − 5x by graph-ing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place. Which function grows more rapidly when x is large?

27. Compare the functions f sxd − x 10 and tsxd − e x by graphing both f and t in several viewing rectangles. When does the graph of t finally surpass the graph of f ?

28. Use a graph to estimate the values of x such that e x . 1,000,000,000.

29. A researcher is trying to determine the doubling time for a population of the bacterium Giardia lamblia. He starts a culture in a nutrient solution and estimates the bacteria count every four hours. His data are shown in the table.

Time (hours) 0 4 8 12 16 20 24

Bacteria countsCFUymLd 37 47 63 78 105 130 173

(a) Make a scatter plot of the data. (b) Use a graphing calculator to find an exponential curve

f std − a ? bt that models the bacteria population t hours later.

;

;

;

;

(c) Graph the model from part (b) together with the scatter plot in part (a). Use the TRACE feature to determine how long it takes for the bacteria count to double.

© S

ebas

tian

Kaul

itzki

/ Sh

utte

rsto

ck.c

om

G. lamblia

30. A bacteria culture starts with 500 bacteria and doubles in size every half hour.

(a) How many bacteria are there after 3 hours? (b) How many bacteria are there after t hours? (c) How many bacteria are there after 40 minutes? (d) Graph the population function and estimate the time

for the population to reach 100,000.

31. The half-life of bismuth-210, 210Bi, is 5 days. (a) If a sample has a mass of 200 mg, find the amount

remaining after 15 days. (b) Find the amount remaining after t days. (c) Estimate the amount remaining after 3 weeks. (d) Use a graph to estimate the time required for the mass

to be reduced to 1 mg.

32. An isotope of sodium, 24Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.

(a) Find the amount remaining after 60 hours. (b) Find the amount remaining after t hours. (c) Estimate the amount remaining after 4 days. (d) Use a graph to estimate the time required for the mass

to be reduced to 0.01 g.

33. Use the graph of V in Figure 11 to estimate the half-life of the viral load of patient 303 during the first month of treatment.

34�. After alcohol is fully absorbed into the body, it is metabo-lized with a half-life of about 1.5 hours. Suppose you have had three alcoholic drinks and an hour later, at midnight, your blood alcohol concentration (BAC) is 0.6 mgymL.

(a) Find an exponential decay model for your BAC t hours after midnight.

(b) Graph your BAC and use the graph to determine when you can drive home if the legal limit is 0.08 mgymL.

Source: Adapted from P. Wilkinson et al., “Pharmaco*kinetics of Ethanol after Oral Administration in the Fasting State,” Journal of Pharmaco*kinetics and Biopharmaceutics 5 (1977): 207–24.

35�. Use a graphing calculator with exponential regression capability to model the population of the world with the

;

;

;

;

;

SeCtION 1.5 Inverse Functions and Logarithms 55

data from 1950 to 2010 in Table 1 on page 49. Use the model to esti mate the population in 1993 and to predict the population in the year 2020.

36. The table gives the population of the United States, in mil-lions, for the years 1900–2010. Use a graphing calculator

Year Population Year Population

1900 76 1960 1791910 92 1970 2031920 106 1980 2271930 123 1990 2501940 131 2000 2811950 150 2010 310

;

with exponential regression capability to model the US population since 1900. Use the model to estimate the population in 1925 and to predict the population in the year 2020.

37. If you graph the function

f sxd −1 2 e 1yx

1 1 e 1yx

you’ll see that f appears to be an odd function. Prove it.

38. Graph several members of the family of functions

f sxd −1

1 1 ae bx

where a . 0. How does the graph change when b changes? How does it change when a changes?

;

;

Table 1 gives data from an experiment in which a bacteria culture started with 100 bac-teria in a limited nutrient medium; the size of the bacteria population was recorded at hourly intervals. The number of bacteria N is a function of the time t: N − f std.

Suppose, however, that the biologist changes her point of view and becomes inter-ested in the time required for the population to reach various levels. In other words, she is thinking of t as a function of N. This function is called the inverse function of f , denoted by f 21, and read “ f inverse.” Thus t − f 21sNd is the time required for the population level to reach N. The values of f 21 can be found by reading Table 1 from right to left or by consulting Table 2. For instance, f 21s550d − 6 because f s6d − 550.

t (hours)

N − f std − population at time t

0 1001 1682 2593 3584 4455 5096 5507 5738 586

Table 1 N as a function of t

N t − f 21 sNd − time to reach N bacteria

100 0168 1259 2358 3445 4509 5550 6573 7586 8

Table 2� t as a function of N

Not all functions possess inverses. Let’s compare the functions f and t whose arrow diagrams are shown in Figure 1. Note that f never takes on the same value twice (any two inputs in A have different outputs), whereas t does take on the same value twice (both 2 and 3 have the same output, 4). In symbols,

ts2d − ts3d

but f sx1 d ± f sx 2 d whenever x1 ± x 2

Functions that share this property with f are called one-to-one functions.

4

3

2

1

10

4

2g

4

3

2

1

10

7

4

2f

FIGURE 1 f is one-to-one; t is not.

56 Chapter 1 Functions and Models

1 Definition A function f is called a one-to-one function if it never takes on the same value twice; that is,

f sx1 d ± f sx2 d whenever x1 ± x2

In the language of inputs and outputs, this definition says that f is one-to-one if each output corresponds to only one input.

If a horizontal line intersects the graph of f in more than one point, then we see from Figure 2 that there are numbers x1 and x2 such that f sx1 d − f sx2 d. This means that f is not one-to-one.

‡fl

y=ƒ

y

x⁄ ¤

Therefore we have the following geometric method for determining whether a func-tion is one-to-one.

Horizontal Line Test A function is one-to-one if and only if no horizontal line intersects its graph more than once.

ExamplE 1 Is the function f sxd − x 3 one-to-one?

SOLUtION 1 If x1 ± x 2, then x 31 ± x 3

2 (two different numbers can’t have the same cube). Therefore, by Definition 1, f sxd − x 3 is one-to-one.

SOLUtION 2 From Figure 3 we see that no horizontal line intersects the graph of f sxd − x 3 more than once. Therefore, by the Horizontal Line Test, f is one-to-one. ■

ExamplE 2 Is the function tsxd − x 2 one-to-one?

SOLUtION 1 This function is not one-to-one because, for instance,

ts1d − 1 − ts21d

and so 1 and 21 have the same output.

SOLUtION 2 From Figure 4 we see that there are horizontal lines that intersect the graph of t more than once. Therefore, by the Horizontal Line Test, t is not one-to-one. ■

One-to-one functions are important because they are precisely the functions that pos-sess inverse functions according to the following definition.

2� Definition Let f be a one-to-one function with domain A and range B.Then its inverse function f 21 has domain B and range A and is defined by

f 21syd − x &? f sxd − y

for any y in B.

FIGURE 2� This function is not one-to-one

because f sx1d − f sx2d.

FIGURE 3 f sxd − x 3 is one-to-one.

y=˛

y

x

FIGURE 4 tsxd − x 2 is not one-to-one.

y=≈

x

y

SeCtION 1.5 Inverse Functions and Logarithms 57

This definition says that if f maps x into y, then f 21 maps y back into x. (If f were not one-to-one, then f 21 would not be uniquely defined.) The arrow diagram in Figure 5 indicates that f 21 reverses the effect of f. Note that

domain of f 21 − range of f

range of f 21 − domain of f

For example, the inverse function of f sxd − x 3 is f 21sxd − x 1y3 because if y − x 3, then

f 21syd − f 21sx 3 d − sx 3 d1y3 − x

CAUTION Do not mistake the 21 in f 21 for an exponent. Thus

f 21sxd does not mean 1

f sxd

The reciprocal 1yf sxd could, however, be written as f f sxdg21.

ExamplE 3 If f s1d − 5, f s3d − 7, and f s8d − 210, find f 21s7d, f 21s5d, and f 21s210d.

SOLUtION From the definition of f 21 we have

f 21s7d − 3 because f s3d − 7

f 21s5d − 1 because f s1d − 5

f 21s210d − 8 because f s8d − 210

The diagram in Figure 6 makes it clear how f 21 reverses the effect of f in this case. ■

The letter x is traditionally used as the independent variable, so when we concentrate on f 21 rather than on f, we usually reverse the roles of x and y in Definition 2 and write

3

f 21sxd − y &? f syd − x

By substituting for y in Definition 2 and substituting for x in (3), we get the follow-ing cancellation equations:

4

f 21s f sxdd − x for every x in A

f s f 21sxdd − x for every x in B

FIGURE 5

x

y

A

B

f – !f

B

5

7

_10

f

A

1

3

8

A

1

3

8

f –!

B

5

7

_10

FIGURE 6 The inverse function reverses inputs and outputs.

58 Chapter 1 Functions and Models

The first cancellation equation says that if we start with x, apply f, and then apply f 21, we arrive back at x, where we started (see the machine diagram in Figure 7). Thus f 21 undoes what f does. The second equation says that f undoes what f 21 does.

x xf ƒ f –!

For example, if f sxd − x 3, then f 21sxd − x 1y3 and so the cancellation equations become

f 21s f sxdd − sx 3 d1y3 − x

f s f 21sxdd − sx 1y3 d3 − x

These equations simply say that the cube function and the cube root function cancel each other when applied in succession.

Now let’s see how to compute inverse functions. If we have a function y − f sxd and are able to solve this equation for x in terms of y, then according to Definition 2 we must have x − f 21syd. If we want to call the independent variable x, we then interchange x and y and arrive at the equation y − f 21sxd.

5 How to Find the Inverse Function of a One-to-One Function f

STEp 1 Write y − f sxd.

STEp 2� Solve this equation for x in terms of y (if possible).

STEp 3 To express f 21 as a function of x, interchange x and y. The resulting equation is y − f 21sxd.

ExamplE 4� Find the inverse function of f sxd − x 3 1 2.

SOLUtION According to (5) we first write

y − x 3 1 2

Then we solve this equation for x:

x 3 − y 2 2

x − s3 y 2 2

Finally, we interchange x and y:

y − s3 x 2 2

Therefore the inverse function is f 21sxd − s3 x 2 2 . ■

The principle of interchanging x and y to find the inverse function also gives us the method for obtaining the graph of f 21 from the graph of f. Since f sad − b if and only if f 21sbd − a, the point sa, bd is on the graph of f if and only if the point sb, ad is on the

FIGURE 7

In Example 4, notice how f 21 reverses the effect of f . The function f is the rule “Cube, then add 2”; f 21 is the rule “Subtract 2, then take the cube root.”

SeCtION 1.5 Inverse Functions and Logarithms 59

graph of f 21. But we get the point sb, ad from sa, bd by reflecting about the line y − x. (See Figure 8.)

y

x

(b, a)

(a, b)

y=x

y

x

f –!

y=x f

FIGURE 8 FIGURE 9

Therefore, as illustrated by Figure 9:

The graph of f 21 is obtained by reflecting the graph of f about the line y − x.

ExamplE 5� Sketch the graphs of f sxd − s21 2 x and its inverse function using the same coordinate axes.

SOLUtION First we sketch the curve y − s21 2 x (the top half of the parabola y 2 − 21 2 x, or x − 2y 2 2 1) and then we reflect about the line y − x to get the graph of f 21. (See Figure 10.) As a check on our graph, notice that the expression for f 21 is f 21sxd − 2x 2 2 1, x > 0. So the graph of f 21 is the right half of the parabola y − 2x 2 2 1 and this seems reasonable from Figure 10. ■

Logarithmic FunctionsIf b . 0 and b ± 1, the exponential function f sxd − bx is either increasing or decreasing and so it is one-to-one by the Horizontal Line Test. It therefore has an inverse function f 21, which is called the logarithmic function with base b and is denoted by logb. If we use the formulation of an inverse function given by (3),

f 21sxd − y &? f syd − x

then we have

6

logb x − y &? by − x

Thus, if x . 0, then logb x is the exponent to which the base b must be raised to give x. For example, log10 0.001 − 23 because 1023 − 0.001.

The cancellation equations (4), when applied to the functions f sxd − bx and f 21sxd − logb x, become

7

logbsbx d − x for every x [ R

blogb x − x for every x . 0

y=xy=ƒ

(0, _1)

y=f –!(x)

(_1, 0)

y

x

FIGURE 10

60 Chapter 1 Functions and Models

The logarithmic function logb has domain s0, `d and range R. Its graph is the reflec-tion of the graph of y − bx about the line y − x.

Figure 11 shows the case where b . 1. (The most important logarithmic functions have base b . 1.) The fact that y − bx is a very rapidly increasing function for x . 0 is reflected in the fact that y − logb x is a very slowly increasing function for x . 1.

Figure 12 shows the graphs of y − logb x with various values of the base b . 1. Since logb 1 − 0, the graphs of all logarithmic functions pass through the point s1, 0d.

y

1

x1

y=log£ x

y=log™ x

y=log∞ xy=log¡¸ x

The following properties of logarithmic functions follow from the corresponding properties of exponential functions given in Section 1.4.

Laws of Logarithms If x and y are positive numbers, then

1. logbsxyd − logb x 1 logb y

2. logbS x

yD − logb x 2 logb y

3. logbsxrd − r logb x (where r is any real number)

ExamplE 6 Use the laws of logarithms to evaluate log2 80 2 log2 5.

SOLUtION Using Law 2, we have

log2 80 2 log2 5 − log2S 80

5 D − log2 16 − 4

because 24 − 16. ■

Natural LogarithmsOf all possible bases b for logarithms, we will see in Chapter 3 that the most convenient choice of a base is the number e, which was defined in Section 1.4. The logarithm with base e is called the natural logarithm and has a special notation:

loge x − ln x

If we put b − e and replace loge with “ln” in (6) and (7), then the defining properties of the natural logarithm function become

FIGURE 11

y=x

y=b®, b>1

y=logb x, b>1

y

x

FIGURE 12�

Notation for logarithmsMost textbooks in calculus and the sciences, as well as calculators, use the notation ln x for the natural logarithm and log x for the “common logarithm,” log10 x. In the more advanced mathematical and scientific literature and in computer languages, however, the notation log x usually denotes the natural logarithm.

SeCtION 1.5 Inverse Functions and Logarithms 61

8

ln x − y &? ey − x

9

lnsex d − x x [ R

e ln x − x x . 0

In particular, if we set x − 1, we get

ln e − 1

ExamplE 7 Find x if ln x − 5.

SOLUtION 1 From (8) we see that

ln x − 5 means e 5 − x

Therefore x − e 5.(If you have trouble working with the “ln” notation, just replace it by loge. Then the

equation becomes loge x − 5; so, by the definition of logarithm, e 5 − x.)

SOLUtION 2 Start with the equation

ln x − 5

and apply the exponential function to both sides of the equation:

e ln x − e 5

But the second cancellation equation in (9) says that e ln x − x. Therefore x − e 5. ■

ExamplE 8 Solve the equation e 523x − 10.

SOLUtION We take natural logarithms of both sides of the equation and use (9):

lnse 523x d − ln 10

5 2 3x − ln 10

3x − 5 2 ln 10

x − 13 s5 2 ln 10d

Since the natural logarithm is found on scientific calculators, we can approximate the solution: to four decimal places, x < 0.8991. ■

ExamplE 9 Express ln a 1 12 ln b as a single logarithm.

SOLUtION Using Laws 3 and 1 of logarithms, we have

ln a 1 12 ln b − ln a 1 ln b 1y2

− ln a 1 lnsb

− ln(asb ) ■

62 Chapter 1 Functions and Models

The following formula shows that logarithms with any base can be expressed in terms of the natural logarithm.

10 Change of Base Formula For any positive number b sb ± 1d, we have

logb x −ln x

ln b

prOOF Let y − logb x. Then, from (6), we have by − x. Taking natural logarithms of both sides of this equation, we get y ln b − ln x. Therefore

y −ln x

ln b ■

Scientific calculators have a key for natural logarithms, so Formula 10 enables us to use a calculator to compute a logarithm with any base (as shown in the following example). Simi larly, Formula 10 allows us to graph any logarithmic function on a graph-ing calculator or computer (see Exercises 43 and 44).

ExamplE 10 Evaluate log8 5 correct to six decimal places.

SOLUtION Formula 10 gives

log8 5 −ln 5

ln 8< 0.773976

■

Graph and Growth of the Natural LogarithmThe graphs of the exponential function y − ex and its inverse function, the natural loga-rithm function, are shown in Figure 13. Because the curve y − ex crosses the y-axis with a slope of 1, it follows that the reflected curve y − ln x crosses the x-axis with a slope of 1.

In common with all other logarithmic functions with base greater than 1, the natural logarithm is an increasing function defined on s0, `d and the y-axis is a vertical asymp-tote. (This means that the values of ln x become very large negative as x approaches 0.)

ExamplE 11 Sketch the graph of the function y − lnsx 2 2d 2 1.

SOLUtION We start with the graph of y − ln x as given in Figure 13. Using the transfor- mations of Section 1.3, we shift it 2 units to the right to get the graph of y − lnsx 2 2d and then we shift it 1 unit downward to get the graph of y − lnsx 2 2d 2 1. (See Fig ure 14.)

y

2 x(3, 0)

x=2

y=ln(x-2)

y

x

y=ln x

(1, 0) 0

y

2 x

x=2

(3, _1)

y=ln(x-2)-1

y

10

x1

y=xy=´

y=ln x

FIGURE 13 The graph of y − ln x is the reflection of the graph of y − ex about the line y − x .

FIGURE 14 ■

SeCtION 1.5 Inverse Functions and Logarithms 63

Although ln x is an increasing function, it grows very slowly when x . 1. In fact, ln x grows more slowly than any positive power of x. To illustrate this fact, we compare approximate values of the functions y − ln x and y − x 1y2 − sx in the following table and we graph them in Figures 15 and 16. You can see that initially the graphs of y − sx and y − ln x grow at comparable rates, but eventually the root function far surpasses the logarithm.

x 1 2 5 10 50 100 500 1000 10,000 100,000

ln x 0 0.69 1.61 2.30 3.91 4.6 6.2 6.9 9.2 11.5

sx 1 1.41 2.24 3.16 7.07 10.0 22.4 31.6 100 316

ln x

sx 0 0.49 0.72 0.73 0.55 0.46 0.28 0.22 0.09 0.04

x0

y

1000

20

y=œ„x

y=ln x

x0

y

1

1

y=œ„x

y=ln x

FIGURE 15 FIGURE 16

Inverse trigonometric FunctionsWhen we try to find the inverse trigonometric functions, we have a slight difficulty: Because the trigonometric functions are not one-to-one, they don’t have inverse func-tions. The difficulty is overcome by restricting the domains of these functions so that they become one-to-one.

You can see from Figure 17 that the sine function y − sin x is not one-to-one (use the Horizontal Line Test). But the function f sxd − sin x, 2�y2 < x < �y2, is one-to-one (see Figure 18). The inverse function of this restricted sine function f exists and is denoted by sin21 or arcsin. It is called the inverse sine function or the arcsine function.

y

0_π π xπ2

y=sin x

y

x

_ π2

π2

FIGURE 18 y − sin x, 2�

2 < x < �2

FIGURE 17

64 Chapter 1 Functions and Models

Since the definition of an inverse function says that

f 21sxd − y &? f syd − x

we have

sin21x − y &? sin y − x and 2�

2< y <

�

2

Thus, if 21 < x < 1, sin21x is the number between 2�y2 and �y2 whose sine is x.

ExamplE 12 Evaluate (a) sin21s12d and (b) tansarcsin 13 d.

SOLUtION

(a) We havesin21s1

2d −�

6

because sins�y6d − 12 and �y6 lies between 2�y2 and �y2.

(b) Let � − arcsin 13, so sin � − 13. Then we can draw a right triangle with angle � as

in Figure 19 and deduce from the Pythagorean Theorem that the third side has length s9 2 1 − 2s2 . This enables us to read from the triangle that

tansarcsin 13 d − tan � −1

2s2 ■

The cancellation equations for inverse functions become, in this case,

sin21ssin xd − x for 2�

2< x <

�

2

sinssin21xd − x for 21 < x < 1

The inverse sine function, sin21, has domain f21, 1g and range f2�y2, �y2g, and its graph, shown in Figure 20, is obtained from that of the restricted sine function (Fig-ure 18) by reflection about the line y − x.

The inverse cosine function is handled similarly. The restricted cosine function f sxd − cos x, 0 < x < �, is one-to-one (see Figure 21) and so it has an inverse function denoted by cos21 or arccos.

cos21x − y &? cos y − x and 0 < y < �

The cancellation equations are

cos21scos xd − x for 0 < x < �

cosscos21xd − x for 21 < x < 1

sin21x ±1

sin x

2 œ„2

3

¨

1

FIGURE 19

y

x1_1

π2

_ π2

FIGURE 2�0 y − sin21 x − arcsin x

FIGURE 2�1 y − cos x, 0 < x < �

y

x

1

ππ2

y

x1

π

_1

π2

SeCtION 1.5 Inverse Functions and Logarithms 65

The inverse cosine function, cos21, has domain f21, 1g and range f0, �g. Its graph is shown in Figure 22.

The tangent function can be made one-to-one by restricting it to the interval s2�y2, �y2d. Thus the inverse tangent function is defined as the inverse of the func-tion f sxd − tan x, 2�y2 , x , �y2. (See Figure 23.) It is denoted by tan21 or arctan.

tan21x − y &? tan y − x and 2�

2, y ,

�

2

ExamplE 13 Simplify the expression cosstan21xd.

SOLUtION 1 Let y − tan21x. Then tan y − x and 2�y2 , y , �y2. We want to find cos y but, since tan y is known, it is easier to find sec y first:

sec2 y − 1 1 tan2 y − 1 1 x 2

sec y − s1 1 x 2 ssince sec y . 0 for 2�y2 , y , �y2d

Thus cosstan21xd − cos y −1

sec y−

1

s1 1 x 2

SOLUtION 2 Instead of using trigonometric identities as in Solution 1, it is perhaps easier to use a diagram. If y − tan21x, then tan y − x, and we can read from Figure 24 (which illustrates the case y . 0) that

cosstan21xd − cos y −1

s1 1 x 2 ■

The inverse tangent function, tan21 − arctan, has domain R and range s2�y2, �y2d. Its graph is shown in Figure 25.

π2

_ π2

y

0x

We know that the lines x − 6�y2 are vertical asymptotes of the graph of tan. Since the graph of tan21 is obtained by reflecting the graph of the restricted tangent function about the line y − x, it follows that the lines y − �y2 and y − 2�y2 are horizontal asymptotes of the graph of tan21.

FIGURE 2�5 y − tan21 x − arctan x

FIGURE 2�2� y − cos21x − arccos x

y

x

1

ππ2

y

x1

π

_1

π2

FIGURE 2�4

œ„„„„„1+≈

1y

x

FIGURE 2�3 y − tan x, 2�

2 , x , �2

π2

π2_

y

0 x

66 Chapter 1 Functions and Models

1. (a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it

is one-to-one?

2. (a) Suppose f is a one-to-one function with domain A and range B. How is the inverse function f 21 defined? What is the domain of f 21? What is the range of f 21?

(b) If you are given a formula for f, how do you find a formula for f 21?

(c) If you are given the graph of f, how do you find the graph of f 21?

3–14� A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

3. x 1 2 3 4 5 6

f sxd 1.5 2.0 3.6 5.3 2.8 2.0

4�. x 1 2 3 4 5 6

f sxd 1.0 1.9 2.8 3.5 3.1 2.9

5�. 6. y

xx

y

y

xx

y

7. 8.

y

xx

y

y

xx

y

9. f sxd − 2x 2 3 10. f sxd − x 4 2 16

11. tsxd − 1 2 sin x 12. tsxd − s3 x

13. f std is the height of a football t seconds after kickoff.

14�. f std is your height at age t.

15�. Assume that f is a one-to-one function. (a) If f s6d − 17, what is f 21s17d? (b) If f 21s3d − 2, what is f s2d?

16. If f sxd − x 5 1 x 3 1 x, find f 21s3d and f s f 21s2dd.

17. If tsxd − 3 1 x 1 e x, find t21s4d.

18. The graph of f is given. (a) Why is f one-to-one? (b) What are the domain and range of f 21? (c) What is the value of f 21s2d? (d) Estimate the value of f 21s0d.

y

x0 1

1

19. The formula C − 59 sF 2 32d, where F > 2459.67,

expresses the Celsius temperature C as a function of the Fahrenheit temperature F. Find a formula for the inverse function and interpret it. What is the domain of the inverse function?

The remaining inverse trigonometric functions are not used as frequently and are summarized here.

11 y − csc21x (| x | > 1) &? csc y − x and y [ s0, �y2g ø s�, 3�y2g

y − sec21x (| x | > 1) &? sec y − x and y [ f0, �y2d ø f�, 3�y2d

y − cot21x sx [ Rd &? cot y − x and y [ s0, �d

The choice of intervals for y in the definitions of csc21 and sec21 is not universally agreed upon. For instance, some authors use y [ f0, �y2d ø s�y2, �g in the definition of sec21. [You can see from the graph of the secant function in Figure 26 that both this choice and the one in (11) will work.]

y

x_1

2ππ

FIGURE 2�6 y − sec x

Section 1.5 Inverse Functions and Logarithms 67

20. In the theory of relativity, the mass of a particle with speed � is

m − f svd −m 0

s1 2 v 2yc 2

where m 0 is the rest mass of the particle and c is the speed of light in a vacuum. Find the inverse function of f and explain its meaning.

21–26 Find a formula for the inverse of the function.

21. f sxd − 1 1 s2 1 3x 22. f sxd −4x 2 1

2x 1 3

23. f sxd − e 2x21 24. y − x 2 2 x, x > 12

25. y − lnsx 1 3d 26. y −1 2 e2x

1 1 e2x

27–28 Find an explicit formula for f 21 and use it to graph f 21, f, and the line y − x on the same screen. To check your work, see whether the graphs of f and f 21 are reflections about the line.

27. f sxd − s4x 1 3 28. f sxd − 1 1 e2x

29–30 Use the given graph of f to sketch the graph of f 21.

29. 30. y

x0 1

1

y

x0 2

1

31. Let f sxd − s1 2 x 2 , 0 < x < 1. (a) Find f 21. How is it related to f ? (b) Identify the graph of f and explain your answer to part (a).

32. Let tsxd − s3 1 2 x 3 . (a) Find t21. How is it related to t? (b) Graph t. How do you explain your answer to part (a)?

33. (a) How is the logarithmic function y − logb x defined? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function

y − logb x if b . 1.

34. (a) What is the natural logarithm? (b) What is the common logarithm? (c) Sketch the graphs of the natural logarithm function and the

natural exponential function with a common set of axes.

35–38 Find the exact value of each expression.

35. (a) log2 32 (b) log8 2

36. (a) log5 1

125 (b) lns1ye 2 d

37. (a) log10 40 1 log10 2.5 (b) log 8 60 2 log 8 3 2 log 8 5

;

;

38. (a) e2ln 2 (b) e lnsln e3d

39–41 Express the given quantity as a single logarithm.

39. ln 10 1 2 ln 5 40. ln b 1 2 ln c 2 3 ln d

41. 13 lnsx 1 2d3 1 1

2 fln x 2 lnsx 2 1 3x 1 2d2g

42. Use Formula 10 to evaluate each logarithm correct to six decimal places.

(a) log5 10 (b) log3 57

43–44 Use Formula 10 to graph the given functions on a common screen. How are these graphs related?

43. y − log1.5 x, y − ln x, y − log10 x, y − log50 x

44. y − ln x, y − log10 x, y − e x, y − 10 x

45. Suppose that the graph of y − log2 x is drawn on a coordi-nate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 ft?

46. Compare the functions f sxd − x 0.1 and tsxd − ln x by graphing both f and t in several viewing rectangles. When does the graph of f finally surpass the graph of t?

47–48 Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3.

47. (a) y − log10sx 1 5d (b) y − 2ln x

48. (a) y − lns2xd (b) y − ln | x |

49–50 (a) What are the domain and range of f ?(b) What is the x-intercept of the graph of f ?(c) Sketch the graph of f.

49. f sxd − ln x 1 2 50. f sxd − lnsx 2 1d 2 1

51–54 Solve each equation for x.

51. (a) e724x − 6 (b) lns3x 2 10d − 2

52. (a) lnsx 2 2 1d − 3 (b) e 2x 2 3e x 1 2 − 0

53. (a) 2x25 − 3 (b) ln x 1 lnsx 2 1d − 1

54. (a) lnsln xd − 1 (b) e ax − Ce bx, where a ± b

55–56 Solve each inequality for x.

55. (a) ln x , 0 (b) e x . 5

56. (a) 1 , e 3x21 , 2 (b) 1 2 2 ln x , 3

57. (a) Find the domain of f sxd − lnse x 2 3d. (b) Find f 21 and its domain.

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;

68 Chapter 1 Functions and Models

5�8. (a) What are the values of e ln 300 and lnse 300d? (b) Use your calculator to evaluate e ln 300 and lnse 300d. What

do you notice? Can you explain why the calculator has trouble?

5�9. Graph the function f sxd − sx 3 1 x 2 1 x 1 1 and explain why it is one-to-one. Then use a computer algebra system to find an explicit expression for f 21sxd. (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)

60. (a) If tsxd − x 6 1 x 4, x > 0, use a computer algebra sys-tem to find an expression for t 21sxd.

(b) Use the expression in part (a) to graph y − tsxd, y − x, and y − t 21sxd on the same screen.

61. If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is n − f std − 100 ∙ 2 ty3.

(a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50,000?

62. When a camera flash goes off, the batteries immediately begin to recharge the flash’s capacitor, which stores electric charge given by

Qstd − Q0s1 2 e 2tya d

(The maximum charge capacity is Q0 and t is measured in seconds.)

(a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90%

of capacity if a − 2?

63–68 Find the exact value of each expression.

63. (a) cos21 s21d (b) sin21s0.5d

64�. (a) tan21 s3 (b) arctans21d

CAS

CAS

65�. (a) csc21 s2 (b) arcsin 1

66. (a) sin21(21ys2 ) (b) cos21(s3 y2) 67. (a) cot21(2s3 ) (b) sec21 2

68. (a) arcsinssins5�y4dd (b) cos(2 sin21 ( 513))

69. Prove that cosssin21 xd − s1 2 x 2 .

70–72 Simplify the expression.

70. tanssin21xd 71. sinstan21xd 72. sins2 arccos xd

73-74� Graph the given functions on the same screen. How are these graphs related?

73. y − sin x, 2�y2 < x < �y2; y − sin21x; y − x

74�. y − tan x, 2�y2 , x , �y2; y − tan21x; y − x

75�. Find the domain and range of the function

tsxd − sin21s3x 1 1d

76. (a) Graph the function f sxd − sinssin21xd and explain the appearance of the graph.

(b) Graph the function tsxd − sin21ssin xd. How do you explain the appearance of this graph?

77. (a) If we shift a curve to the left, what happens to its reflection about the line y − x? In view of this geo-metric principle, find an expression for the inverse of tsxd − f sx 1 cd, where f is a one-to-one function.

(b) Find an expression for the inverse of hsxd − f scxd, where c ± 0.

;

;

CONCEpT CHECK Answers to the Concept Check can be found on the back endpapers.

1 REvIEw

1. (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of

a function?

2. Discuss four ways of representing a function. Illustrate your discussion with examples.

3. (a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function.

(b) What is an odd function? How can you tell if a function is odd by looking at its graph? Give three examples of an odd function.

4�. What is an increasing function?

5�. What is a mathematical model?

6. Give an example of each type of function. (a) Linear function (b) Power function (c) Exponential function (d) Quadratic function (e) Polynomial of degree 5 (f ) Rational function

7. Sketch by hand, on the same axes, the graphs of the following functions.

(a) f sxd − x (b) tsxd − x 2

(c) hsxd − x 3 (d) jsxd − x 4

Chapter 1 Review 69

TRUE-FaLSE qUIz

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.

1. If f is a function, then f ss 1 td − f ssd 1 f std.

2. If f ssd − f std, then s − t.

3. If f is a function, then f s3xd − 3 f sxd.

4�. If x1 , x2 and f is a decreasing function, then f sx1 d . f sx2 d

5�. A vertical line intersects the graph of a function at most once.

6. If f and t are functions, then f 8 t − t 8 f.

7. If f is one-to-one, then f 21sxd −1

f sxd.

8. You can always divide by e x.

9. If 0 , a , b, then ln a , ln b.

10. If x . 0, then sln xd6 − 6 ln x.

11. If x . 0 and a . 1, then ln x

ln a− ln

x

a .

12. tan21s21d − 3�y4

13. tan21x −sin21x

cos21x 14�. If x is any real number, then sx 2 − x.

8. Draw, by hand, a rough sketch of the graph of each function. (a) y − sin x (b) y − tan x (c) y − e x

(d) y − ln x (e) y − 1yx (f ) y − | x | (g) y − sx (h) y − tan21x

9. Suppose that f has domain A and t has domain B. (a) What is the domain of f 1 t? (b) What is the domain of f t? (c) What is the domain of fyt?

10. How is the composite function f 8 t defined? What is its domain?

11. Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows.

(a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reflect about the x-axis.

(f ) Reflect about the y-axis. (g) Stretch vertically by a factor of 2. (h) Shrink vertically by a factor of 2. (i) Stretch horizontally by a factor of 2. ( j) Shrink horizontally by a factor of 2.

12. (a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph?

(b) If f is a one-to-one function, how is its inverse function f 21 defined? How do you obtain the graph of f 21 from the graph of f ?

13. (a) How is the inverse sine function f sxd − sin21x defined? What are its domain and range?

(b) How is the inverse cosine function f sxd − cos21x defined? What are its domain and range?

(c) How is the inverse tangent function f sxd − tan21x defined? What are its domain and range?

(f) Is f one-to-one? Explain. (g) Is f even, odd, or neither even nor odd? Explain.

2. The graph of t is given.

gy

x0 1

1

(a) State the value of ts2d. (b) Why is t one-to-one? (c) Estimate the value of t21s2d. (d) Estimate the domain of t21. (e) Sketch the graph of t21.

EXERCISES

1. Let f be the function whose graph is given.

y

x1

1

f

(a) Estimate the value of f s2d. (b) Estimate the values of x such that f sxd − 3. (c) State the domain of f. (d) State the range of f. (e) On what interval is f increasing?

70 Chapter 1 Functions and Models

21. Life expectancy improved dramatically in the 20th century. The table gives the life expectancy at birth (in years) of males born in the United States. Use a scatter plot to choose an appropriate type of model. Use your model to predict the life span of a male born in the year 2010.

Birth year Life expectancy Birth year Life expectancy

1900 48.3 1960 66.61910 51.1 1970 67.11920 55.2 1980 70.01930 57.4 1990 71.81940 62.5 2000 73.01950 65.6

22. A small-appliance manufacturer finds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week.

(a) Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then sketch the graph.

(b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it

represent?

23. If f sxd − 2x 1 ln x, find f 21s2d.

24�. Find the inverse function of f sxd −x 1 1

2x 1 1.

25�. Find the exact value of each expression. (a) e 2 ln 3 (b) log10 25 1 log10 4

(c) tansarcsin 12 d (d) sinscos21 s45dd

26. Solve each equation for x. (a) e x − 5 (b) ln x − 2

(c) eex− 2 (d) tan21x − 1

27. The half-life of palladium-100, 100Pd, is four days. (So half of any given quantity of 100Pd will disintegrate in four days.) The initial mass of a sample is one gram.

(a) Find the mass that remains after 16 days. (b) Find the mass mstd that remains after t days. (c) Find the inverse of this function and explain its meaning. (d) When will the mass be reduced to 0.01g?

28. The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is

Pstd −100,000

100 1 900e2t

where t is measured in years. (a) Graph this function and estimate how long it takes for the

population to reach 900. (b) Find the inverse of this function and explain its meaning. (c) Use the inverse function to find the time required for

the population to reach 900. Compare with the result of part (a).

;

3. If f sxd − x 2 2 2x 1 3, evaluate the difference quotient

f sa 1 hd 2 f sadh

4�. Sketch a rough graph of the yield of a crop as a function of the amount of fertilizer used.

5�–8 Find the domain and range of the function. Write your answer in interval notation.

5�. f sxd − 2ys3x 2 1d 6. tsxd − s16 2 x 4

7. hsxd − lnsx 1 6d 8. Fstd − 3 1 cos 2t

9. Suppose that the graph of f is given. Describe how the graphs of the following functions can be obtained from the graph of f.

(a) y − f sxd 1 8 (b) y − f sx 1 8d (c) y − 1 1 2 f sxd (d) y − f sx 2 2d 2 2 (e) y − 2f sxd (f ) y − f 21sxd

10. The graph of f is given. Draw the graphs of the following functions.

(a) y − f sx 2 8d (b) y − 2f sxd (c) y − 2 2 f sxd (d) y − 1

2 f sxd 2 1 (e) y − f 21sxd (f ) y − f 21sx 1 3d

y

x0 1

1

11–16 Use transformations to sketch the graph of the function.

11. y − sx 2 2d3 12. y − 2sx

13. y − x 2 2 2x 1 2 14�. y − lnsx 1 1d

15�. f sxd − 2cos 2x 16. f sxd − H2x

e x 2 1

if x , 0

if x > 0

17. Determine whether f is even, odd, or neither even nor odd.

(a) f sxd − 2x 5 2 3x 2 1 2

(b) f sxd − x 3 2 x 7

(c) f sxd − e2x2

(d) f sxd − 1 1 sin x

18. Find an expression for the function whose graph consists of the line segment from the point s22, 2d to the point s21, 0d together with the top half of the circle with center the origin and radius 1.

19. If f sxd − ln x and tsxd − x 2 2 9, find the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , (d) t 8 t, and their domains.

20. Express the function Fsxd − 1ysx 1 sx as a composition of three functions.

71

principles of problem Solving

There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to give some principles that may be useful in the solution of certain problems. These steps and principles are just common sense made explicit. They have been adapted from George Polya’s book How To Solve It.

The first step is to read the problem and make sure that you understand it clearly. Ask yourself the following questions:

What is the unknown?

What are the given quantities?

What are the given conditions?

For many problems it is useful to

draw a diagram

and identify the given and required quantities on the diagram.Usually it is necessary to

introduce suitable notation

In choosing symbols for the unknown quantities we often use letters such as a, b, c, m, n, x, and y, but in some cases it helps to use initials as suggestive symbols; for instance, V for volume or t for time.

Find a connection between the given information and the unknown that will enable you to calculate the unknown. It often helps to ask yourself explicitly: “How can I relate the given to the unknown?” If you don’t see a connection immediately, the following ideas may be helpful in devising a plan.

Try to Recognize Something Familiar Relate the given situation to previous knowledge. Look at the unknown and try to recall a more familiar problem that has a similar unknown.

Try to Recognize Patterns Some problems are solved by recognizing that some kind of pattern is occurring. The pattern could be geometric, or numerical, or algebraic. If you can see regularity or repetition in a problem, you might be able to guess what the con-tinuing pattern is and then prove it.

Use Analogy Try to think of an analogous problem, that is, a similar problem, a related problem, but one that is easier than the original problem. If you can solve the similar, simpler problem, then it might give you the clues you need to solve the original, more difficult problem. For instance, if a problem involves very large numbers, you could first try a similar problem with smaller numbers. Or if the problem involves three-dimensional geometry, you could look for a similar problem in two-dimensional geometry. Or if the problem you start with is a general one, you could first try a special case.

Introduce Something Extra It may sometimes be necessary to introduce something new, an auxiliary aid, to help make the connection between the given and the unknown. For instance, in a problem where a diagram is useful the auxiliary aid could be a new line drawn in a diagram. In a more algebraic problem it could be a new unknown that is related to the original unknown.

Take Cases We may sometimes have to split a problem into several cases and give a different argument for each of the cases. For instance, we often have to use this strategy in dealing with absolute value.

1 UndERSTAnd ThE PRoblEm

2 ThInk oF A PlAn

72

Work backward Sometimes it is useful to imagine that your problem is solved and work backward, step by step, until you arrive at the given data. Then you may be able to reverse your steps and thereby construct a solution to the original problem. This pro-cedure is commonly used in solving equations. For instance, in solving the equation 3x 2 5 − 7, we suppose that x is a number that satisfies 3x 2 5 − 7 and work back-ward. We add 5 to each side of the equation and then divide each side by 3 to get x − 4. Since each of these steps can be reversed, we have solved the problem.

Establish Subgoals In a complex problem it is often useful to set subgoals (in which the desired situation is only partially fulfilled). If we can first reach these subgoals, then we may be able to build on them to reach our final goal.

Indirect Reasoning Sometimes it is appropriate to attack a problem indirectly. In using proof by contradiction to prove that P implies Q, we assume that P is true and Q is false and try to see why this can’t happen. Somehow we have to use this information and arrive at a contradiction to what we absolutely know is true.

mathematical Induction In proving statements that involve a positive integer n, it is frequently helpful to use the following principle.

principle of Mathematical Induction Let Sn be a statement about the positive integer n. Suppose that

1. S1 is true.

2. Sk11 is true whenever Sk is true.

Then Sn is true for all positive integers n.

This is reasonable because, since S1 is true, it follows from condition 2 swith k − 1d that S2 is true. Then, using condition 2 with k − 2, we see that S3 is true. Again using condition 2, this time with k − 3, we have that S4 is true. This procedure can be followed indefinitely.

In Step 2 a plan was devised. In carrying out that plan we have to check each stage of the plan and write the details that prove that each stage is correct.

Having completed our solution, it is wise to look back over it, partly to see if we have made errors in the solution and partly to see if we can think of an easier way to solve the problem. Another reason for looking back is that it will familiarize us with the method of solution and this may be useful for solving a future problem. Descartes said, “Every problem that I solved became a rule which served afterwards to solve other problems.”

These principles of problem solving are illustrated in the following examples. Before you look at the solutions, try to solve these problems yourself, referring to these Principles of Problem Solving if you get stuck. You may find it useful to refer to this section from time to time as you solve the exercises in the remaining chapters of this book.

ExamplE 1 Express the hypotenuse h of a right triangle with area 25 m2 as a function of its perimeter P.

SOLUtION Let’s first sort out the information by identifying the unknown quantity and the data:

Unknown: hypotenuse h

Given quantities: perimeter P, area 25 m 2

3 CARRy oUT ThE PlAn

4� look bACk

PS Understand the problem

73

It helps to draw a diagram and we do so in Figure 1.

a

hb

In order to connect the given quantities to the unknown, we introduce two extra variables a and b, which are the lengths of the other two sides of the triangle. This enables us to express the given condition, which is that the triangle is right-angled, by the Pythago rean Theorem:

h 2 − a 2 1 b 2

The other connections among the variables come by writing expressions for the area and perimeter:

25 − 12 ab P − a 1 b 1 h

Since P is given, notice that we now have three equations in the three unknowns a, b, and h:

1 h 2 − a 2 1 b 2

2� 25 − 12 ab

3 P − a 1 b 1 h

Although we have the correct number of equations, they are not easy to solve in a straightforward fashion. But if we use the problem-solving strategy of trying to recog-nize something familiar, then we can solve these equations by an easier method. Look at the right sides of Equations 1, 2, and 3. Do these expressions remind you of anything familiar? Notice that they contain the ingredients of a familiar formula:

sa 1 bd2 − a 2 1 2ab 1 b 2

Using this idea, we express sa 1 bd2 in two ways. From Equations 1 and 2 we have

sa 1 bd2 − sa 2 1 b 2 d 1 2ab − h 2 1 4s25d

From Equation 3 we have

sa 1 bd2 − sP 2 hd2 − P2 2 2Ph 1 h 2

Thus h 2 1 100 − P2 2 2Ph 1 h 2

2Ph − P2 2 100

h −P2 2 100

2P

This is the required expression for h as a function of P. ■

As the next example illustrates, it is often necessary to use the problem-solving prin-ciple of taking cases when dealing with absolute values.

ExamplE 2 Solve the inequality | x 2 3 | 1 | x 1 2 | , 11.

SOLUtION Recall the definition of absolute value:

| x | − Hx

2x

if x > 0

if x , 0

PS Draw a diagram

FIGURE 1

PS Connect the given with the unknownPS Introduce something extra

PS Relate to the familiar

74

It follows that

| x 2 3 | − Hx 2 3

2sx 2 3dif x 2 3 > 0

if x 2 3 , 0

− Hx 2 3

2x 1 3

if x > 3

if x , 3

Similarly

| x 1 2 | − Hx 1 2

2sx 1 2dif x 1 2 > 0

if x 1 2 , 0

− Hx 1 2

2x 2 2

if x > 22

if x , 22

These expressions show that we must consider three cases:

x , 22 22 < x , 3 x > 3

CaSe I If x , 22, we have

| x 2 3 | 1 | x 1 2 | , 11

2x 1 3 2 x 2 2 , 11

22x , 10

x . 25

CaSe II If 22 < x , 3, the given inequality becomes

2x 1 3 1 x 1 2 , 11

5 , 11 (always true)

CaSe III If x > 3, the inequality becomes

x 2 3 1 x 1 2 , 11

2x , 12

x , 6

Combining cases I, II, and III, we see that the inequality is satisfied when 25 , x , 6. So the solution is the interval s25, 6d. ■

In the following example we first guess the answer by looking at special cases and recognizing a pattern. Then we prove our conjecture by mathematical induction.

In using the Principle of Mathematical Induction, we follow three steps:

Step 1 Prove that Sn is true when n − 1.

Step 2 Assume that Sn is true when n − k and deduce that Sn is true when n − k 1 1.

Step 3 Conclude that Sn is true for all n by the Principle of Mathematical Induction.

PS Take cases

75

ExamplE 3 If f0sxd − xysx 1 1d and fn11 − f0 8 fn for n − 0, 1, 2, . . . , find a formula for fnsxd.

SOLUtION We start by finding formulas for fnsxd for the special cases n − 1, 2, and 3.

f1sxd − s f0 8 f0dsxd − f0( f0sxd) − f0S x

x 1 1D −

x

x 1 1

x

x 1 11 1

−

x

x 1 1

2x 1 1

x 1 1

−x

2x 1 1

f2sxd − s f0 8 f1 dsxd − f0( f1sxd) − f0S x

2x 1 1D −

x

2x 1 1

x

2x 1 11 1

−

x

2x 1 1

3x 1 1

2x 1 1

−x

3x 1 1

f3sxd − s f0 8 f2 dsxd − f0( f2sxd) − f0S x

3x 1 1D −

x

3x 1 1

x

3x 1 11 1

−

x

3x 1 1

4x 1 1

3x 1 1

−x

4x 1 1

We notice a pattern: The coefficient of x in the denominator of fnsxd is n 1 1 in the three cases we have computed. So we make the guess that, in general,

4 fnsxd −x

sn 1 1dx 1 1

To prove this, we use the Principle of Mathematical Induction. We have already verified that (4) is true for n − 1. Assume that it is true for n − k, that is,

fksxd −x

sk 1 1dx 1 1

Then fk11sxd − s f0 8 fk dsxd − f0( fksxd) − f0S x

sk 1 1dx 1 1D −

x

sk 1 1dx 1 1

x

sk 1 1dx 1 11 1

−

x

sk 1 1dx 1 1

sk 1 2dx 1 1

sk 1 1dx 1 1

−x

sk 1 2dx 1 1

This expression shows that (4) is true for n − k 1 1. Therefore, by mathematical induction, it is true for all positive integers n. ■

PS Analogy: Try a similar, simpler problem

PS Look for a pattern

76

1. One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpen-dicular to the hypotenuse as a function of the length of the hypotenuse.

2. The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter.

3. Solve the equation | 2x 2 1 | 2 | x 1 5 | − 3.

4�. Solve the inequality | x 2 1 | 2 | x 2 3 | > 5.

5�. Sketch the graph of the function f sxd − | x 2 2 4 | x | 1 3 |. 6. Sketch the graph of the function tsxd − | x 2 2 1 | 2 | x 2 2 4 |. 7. Draw the graph of the equation x 1 | x | − y 1 | y |. 8. Sketch the region in the plane consisting of all points sx, yd such that

| x 2 y | 1 | x | 2 | y | < 2

9. The notation maxha, b, . . .j means the largest of the numbers a, b, . . . . Sketch the graph of each function.

(a) f sxd − maxhx, 1yxj (b) f sxd − maxhsin x, cos xj (c) f sxd − maxhx 2, 2 1 x, 2 2 xj

10. Sketch the region in the plane defined by each of the following equations or inequalities. (a) maxhx, 2yj − 1

(b) 21 < maxhx, 2yj < 1

(c) maxhx, y 2j − 1

11. Evaluate slog2 3dslog3 4dslog4 5d ∙ ∙ ∙ slog31 32d.

12. (a) Show that the function f sxd − ln(x 1 sx 2 1 1 ) is an odd function.

(b) Find the inverse function of f.

13. Solve the inequality lnsx 2 2 2x 2 2d < 0.

14�. Use indirect reasoning to prove that log2 5 is an irrational number.

15�. A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of 30 miyh; she drives the second half at 60 miyh. What is her average speed on this trip?

16. Is it true that f 8 st 1 hd − f 8 t 1 f 8 h?

17. Prove that if n is a positive integer, then 7n 2 1 is divisible by 6.

18. Prove that 1 1 3 1 5 1 ∙ ∙ ∙ 1 s2n 2 1d − n2.

19. If f0sxd − x 2 and fn11sxd − f0s fnsxdd for n − 0, 1, 2, . . . , find a formula for fnsxd.

20. (a) If f0sxd −1

2 2 x and fn11 − f0 8 fn for n − 0, 1, 2, . . . , find an expression for fnsxd and

use mathematical induction to prove it.

(b) Graph f0, f1, f2, f3 on the same screen and describe the effects of repeated composition.;

problems