Coburn J.W., Herdlick J.D. College Algebra: Graphs and Models

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350 CHAPTER 3 Quadratic Functions and Operations on Functions 3–70

College Algebra G&M—

䊳

APPLICATIONS

71. Boat manufacturing: Giaro Boats manufactures a

popular recreational vessel, the Revolution. To plan

for expanded production and increased labor costs,

the company carefully tracks current costs and

income. The ﬁxed cost to produce this boat is

$108,000 and the variable costs are $28,000 per

boat. If the Revolution sells for $40,000, (a) ﬁnd

the proﬁt function and (b) determine how many

boats must be sold for the company to break even.

72. Non-proﬁt publications: Adobe Hope, a nonproﬁt

agency, publishes the weekly newsletter

Community Options. In doing so, they provide

useful information to the surrounding area while

giving high school dropouts valuable work

experience. The ﬁxed cost for publishing the

newsletter is $900 per week, with a variable cost of

$0.25 per newsletter. If the newsletter is sold for

$1.50 per copy, (a) ﬁnd the proﬁt function for the

newsletter, (b) determine how many newsletters

must be sold to break even, and (c) determine how

much money will be returned to the community if

1000 newsletters are sold (to preserve their status

as a nonproﬁt organization).

73. Cost, revenue, and proﬁt: Suppose the total cost of

manufacturing a certain computer component can be

modeled by the function where n is

the number of components made and C(n) is in

dollars. If each component is sold at a price of

$11.45, the revenue is modeled by

Use this information to complete the following.

a. Find the function that represents the total proﬁt

made from sales of the components.

b. How much proﬁt is earned if 12 components

are made and sold?

c. How much proﬁt is earned if 60 components

are made and sold?

d. Explain why the company is making a

“negative proﬁt” after the 114th component is

made and sold.

74. Cost, revenue, and proﬁt: For a certain

manufacturer, revenue has been increasing but so

has the cost of materials and the cost of employee

beneﬁts. Suppose revenue can be modeled by

the cost of materials by

and the cost of beneﬁts by

where t represents the number of

months since operations began and outputs are in

thousands of dollars. Use this information to

complete the following.

a. Find the function that represents the total

manufacturing costs.

B1t2 0.1t

 2,

M1t2 2t  1,

R1t2 101t

R1n2 11.45n.

C1n2 0.1n

b. Find the function that represents how much

more the cost of beneﬁts are than the cost of

materials.

c. What was the cost of beneﬁts in the 10th

month after operations began?

d. How much less were the beneﬁts costs than the

cost of materials in the 10th month?

e. Find the function that represents the proﬁt earned

by this company.

f. Find the amount of proﬁt earned in the 5th

month and 10th month. Discuss each result.

Minimizing travel distance is one of the main

responsibilities of a logistics manager. From delivering

pizzas on a Friday night, to building distribution centers

for goods and commodities, a shorter total travel

distance translates directly into reduced costs. On a

straight highway, the distance from a distribution center

at milepost x to a distribution point at milepost m

. Using the addition of functions, the total

distance from a distribution center to its n distribution

points is .

75. Minimizing CNG distribution distance: Ruta 2

in Argentina is a well-traveled corridor for

residents of Buenos Aires vacationing in Mar del

Plata. Many of the cars in Argentina operate on

compressed natural gas (CNG). Suppose there

were CNG ﬁlling stations at kilometer-posts 30,

49, 115, 199, and 280. Using a graphing calculator,

at what kilometer-post should the CNG storage

tanks supplying these stations be built in order to

minimize total delivery distance?

76. Minimizing H

O distribution distance: The Trans-

Australian Railway links Australia’s eastern states

and those of Western Australia. Most of the rail line

lies in the central desert (with summer temperatures

of 50C/122F), which is home to the world’s longest

stretch of dead-straight track, over 478 km/297 mi.

Assume rest stops are scheduled to be built at mile-

posts 25, 56, 89, 145, and 229 miles from the east. At

what mile-post should the water storage tanks

supplying these rest stops be built in order to

minimize the total delivery distance?

77. Minimizing CNG distribution distance: In

addition to the ﬁve already mentioned in Exercise 75,

another CNG station is being constructed at

kilometer-post 233 on Ruta 2. With this station

included, where should the CNG storage tanks be

built? How does this result compare with that of

Exercise 75?

D1x2ⴝ 円x ⴚ m

円 ⴙ 円x ⴚ m

円 ⴙ

ⴙ 円x  m

円

d ⴝ 円x ⴚ m

円

cob19545_ch03_340-352.qxd 11/25/10 3:43 PM Page 350

. (a) If the well is 230 ft deep,

what is the total time? (b) If it takes 6 sec for the

sound to be heard, how deep is the well?

80. Predator-prey concentrations: Suppose the

monthly whitetip reef shark population off the

coast of Manuel Antonio National Park (Costa

Rica) can be modeled by the function

(where

corresponds to January). One of its favorite foods

is the spiny lobster, whose monthly population in

these waters might be modeled by the function

. The number

of lobsters per shark in any given month

can then be modeled by the quotient

. (a) How many

lobsters per shark are there in February? (b) When

is the number of lobsters per shark a minimum?

What is this minimum?

N1t2

110t

 1500t  13,400

40t

 500t  1300

l1t2 110t

 1500t  13,400

t  1s1t240t

 500t  1300

T1d2



1116

78. Minimizing H

O distribution distance: In

addition to the ﬁve already mentioned in Exercise 76,

another rest stop is scheduled to be built at mile-

post 192 on the Trans-Australian Railway. With

this rest stop included, where should the water

storage tanks be built? How does this result

compare with that of Exercise 76?

79. Measuring the depth of a well: To estimate the

depth of a well, a large stone is dropped into the

opening and a stop-watch is used to measure the

number of seconds until a splash is heard. By

solving the freefall equation for t, we ﬁnd

the function models the time it takes

for the stone to hit the water. Since the speed of

sound is known to be 1116 ft/sec, we use the

distance  rate  time equation to obtain

as a function modeling the time it takes

the sound of the splash to reach our ears. The total

time is the sum of these two functions, let’s call it

1d2

1116

1d2

d  16t

3–71 Section 3.5 The Algebra of Functions 351

College Algebra G&M—

䊳

EXTENDING THE CONCEPT

81. For a certain country, assume the function

approximates the number of

Conservatives in the senate for the years 1995 to

2007, where corresponds to 1995. The function

gives the number of Liberals for

these years. Use this information to answer the

following. (a) During what years did the

Conservatives control the senate? (b) What was the

greatest difference between the number of seats

held by each faction in any one year? In what year

did this occur? (c) What was the minimum number

of seats held by the Conservatives? In what year?

(d) Assuming no independent or third-party

candidates are elected, what information does the

function give us? What

information does give us?

82. Given and , what

can you say about the domain of ? Enter

the functions as Y

and Y

on a graphing calculator,

then enter See if you can

determine why the calculator gives an error

message for Y

, regardless of the input.

83. Use a graphing calculator to ﬁnd the zeroes of each

new function in Exercises 67 and 68, and describe

their signiﬁcance.

 Y

 Y

1f  g21x2

g1x2 1x  2

f 1x2 11  x

t1x2

冟

C1x2 L1x2

冟

T1x2 C1x2 L1x2

21.7215x  10

L1x20.0345x

 0.8996x

 7.5383x



x  0

21.7215x  40

C1x2 0.0345x

 0.8996x

 7.5383x



84. Instead of calculating the result of an operation on

two functions at a speciﬁc point as in Exercises

47–52, we can actually graph the function that

results from the operation. This skill, called the

addition of ordinates, is widely applied in a study

of tides and other areas. For

and , complete a table of values

like the one shown for . For the last

column, remember that ,

and use this relation to complete the column.

Finally, use the ordered pairs to

graph the new function. Is the new function

smooth? Is the new function continuous?

1x, 1f  g21x22

1f  g21x2 f

1x2 g1x2

x 僆 32, 84

g1x2 4

冟

x  3

冟

 5

1x2 1x  32

 2

xf(x) g(x)(f ⴚ g)(x)

1

2

Exercise 84

cob19545_ch03_340-352.qxd 8/10/10 6:29 PM Page 351

352 CHAPTER 3 Quadratic Functions and Operations on Functions 3–72

College Algebra G&M—

䊳

MAINTAINING YOUR SKILLS

85. (R.3) Solve the equation.

86. (2.2) Given , ﬁnd and

describe the resulting transformation.

87. (1.1/1.2) Given the points and ,

ﬁnd (a) the standard form of the equation of the

line containing these points and (b) the distance

between these points.

13, 1212, 52

1x  12f 1x2 3x

 2x

1x  32



x  1

88. (3.4) Find the average rate of change of the

function over the indicated

intervals.

a. [1, 2]

b. [1, 3]

c. [1, 4]

g1x2 x

 9x  1

3.6 The Composition of Functions and the Difference Quotient

The composition of functions gives us an efﬁcient

way to study how relationships are “linked.” For

example, the number of wolves w in a countywide

area depends on the human population x, and a

simple model might be . But

the number of rodents r depends on the number of

wolves, say , so the human

population also has a measurable effect on the rodent

population. In this section, we’ll show that this effect

is modeled by the function .

A. The Composition of Functions

The composition of functions is best understood by studying the “input/output” nature

of a function. Consider For g(x) we might say, “inputs are squared,

then decreased by three.” In diagram form we have:

g1x2 x

 3.

r1x2 4300  0.19x

r1w2 10,000  9.5w

w1x2 600  0.02x

LEARNING OBJECTIVES

In Section 3.6 you will see how we can:

A. Compose two functions

and determine the domain;

decompose a function

B. Interpret the composition

of functions numerically

and graphically

C. Apply the difference

quotient to find the

average rate of change

for nonlinear functions

D. Apply the composition of

functions and the

difference quotient in

context

g(x)

inputs are squared,

then decreased by three

Output

(input)

 3

Input

(input)

ⴚ 3

In many respects, a function box can be regarded as a very simple machine, run-

ning a simple program. It doesn’t matter what the input is, this machine is going to

square the input then subtract three.

cob19545_ch03_340-352.qxd 11/25/10 3:44 PM Page 352

College Algebra G&M—College Algebra G&M—

EXAMPLE 1

䊳

Evaluating a Function

For ﬁnd

Solution

䊳

a. original function

input

square input, then subtract 3

simplify

result

b. original function

input 5

square input, then subtract 3

result

c. original function

input

square input, then subtract 3

expand binomial

result

Now try Exercises 7 and 8

䊳

When the input value is itself a function (rather than a single number or variable),

this process is called the composition of functions. The evaluation method is exactly

the same, we are simply using a function input. Using a general function g(x) and a

function diagram as before, we illustrate the process in Figure 3.70.

⫽ t

⫺ 8t ⫹ 13

⫽ t

⫺ 8t ⫹ 16 ⫺ 3

g1t ⫺ 42⫽ 1t ⫺ 42

⫺ 3

t ⫺ 4

g1x2⫽ x

⫺ 3

⫽ 25t

⫺ 3

g15t2⫽ 15t2

⫺ 3

g1x2⫽ x

⫺ 3

⫽ 22

⫽ 25 ⫺ 3

g1⫺52⫽ 1⫺52

⫺ 3

⫺5

g1x2⫽ x

⫺ 3

g1t ⫺ 42

g15t2

g1⫺52

g1x2⫽ x

⫺ 3,

3–73 Section 3.6 The Composition of Functions and the Difference Quotient 353

Input g(x)

g(x)

Output

g(x)

Input x

f(x)

Output ( f

⬚

g)(x) = f [g(x)]

g specifies

operations on x

f specifies

operations on

g(x)

Figure 3.70

WORTHY OF NOTE

It’s important to note that t and

are two different, distinct

values—the number represented

by t, and a number four less than t.

Examples would be 7 and 3, 12

and 8, as well as ⫺10 and ⫺14.

There should be nothing awkward

or unusual about evaluating g(t)

versus evaluating as in

Example 1(c).

g1t ⫺ 42

t ⫺ 4

The notation used for the composition of f with g is an open dot “ ” placed be-

tween them, and is read, “f composed with g.” The notation indicates that

g(x) is an input for f: If the order is reversed, as in

becomes the input for g: Figure 3.70 also helps us determine the

domain of a composite function, in that the ﬁrst function g can operate only if x is a

valid input for g, and the second function f can operate only if g(x) is a valid input for f.

In other words, is deﬁned for all x in the domain of g, such that g(x) is in the

domain of f.

1f ⴰ g21x2

1g ⴰ f

21x2⫽ g3f 1x24.

1g ⴰ f 21x2, f

1x21 f ⴰ g21x2⫽ f 3g1x24.

1f ⴰ g21x2

ⴰ

CAUTION

䊳

Try not to confuse the new “open dot” notation for the composition of functions, with the

multiplication dot used to indicate the product of two functions: while

.1f ⴰ g21x2⫽ f 3g1x24

g21x2⫽ 1fg21x2

cob19545_ch03_353-370.qxd 8/10/10 7:49 PM Page 353

WORTHY OF NOTE

Example 2 shows that is

generally not equal to . On

those occasions when they are

equal, the functions have a unique

relationship that we’ll study in

Section 5.1.

1g ⴰ f 21x2

1f ⴰ g21x2

354 CHAPTER 3 Quadratic Functions and Operations on Functions 3–74

College Algebra G&M—

Figure 3.71

The Composition of Functions

Given two functions f and g, the composition of f with g is deﬁned by

The domain of the composition is all x in the domain of g

for which g(x) is in the domain of f.

In Figure 3.71, these ideas are displayed using mapping notation, as we consider

the simple case where and .f

1x2⫽ 1xg1x2⫽ x

1f ⴰ g21x2⫽ f 3g1x24

The domain of g (all real numbers) is shown within the red border, with g taking

the negative inputs represented by x

(light red), to a like-colored portion of the

range—the negative outputs g(x

). The nonnegative inputs represented by x

(dark

red) are also mapped to a like-colored portion of the range—the nonnegative outputs

g(x

). While the range of g is also all real numbers, function f can only use the nonneg-

ative inputs represented by g(x

). This restricts the domain of to only the

inputs from g, where g(x) is in the domain of f.

EXAMPLE 2

䊳

Finding a Composition of Functions

Given and ﬁnd

Also determine the domain for each.

Solution

䊳

a. says “decrease inputs by 4, and take the square root of the

result.”

(

) is an input for

decrease input by 4, and take the square root of the result

substitute for

(

)

result

While g is deﬁned for all real numbers, f(x) represents a real number only

when . For , this means we need , giving ,

. In interval notation, the domain of is .

b. The function g says “inputs are multiplied by 3, then increased by 2.”

(

) is an input for

multiply input by 3, then increase by 2

substitute for

(

)

For g[ f(x)], g can accept any real number input, but f can supply only those

where The domain of is

Now try Exercises 9 through 18

䊳

x 僆 34, q2.1g ⴰ f21x2x ⱖ 4.

1x ⫺ 4 ⫽ 31x ⫺ 4 ⫹ 2

⫽ 3f

1x2⫹ 2

1g ⴰ f 21x2⫽ g3f

1x24

x 僆 3

, q21f ⴰ g21x2x ⱖ

3x ⫹ 2 ⱖ 4g1x2ⱖ 4f 3g1x24x ⱖ 4

⫽ 13x ⫺ 2

3x ⫹ 2 ⫽ 113x ⫹ 22⫺ 4

⫽ 1g1x2⫺ 4

1f ⴰ g21x2⫽ f 3g1x24

1x2⫽ 1x ⫺ 4

1g ⴰ f 21x2

1f ⴰ g21x2

g1x2⫽ 3x ⫹ 2,f

1x2⫽ 1x ⫺ 4

1f ⴰ g21x2

Domain of f

⬚

Domain of g

Domain of fRange of g

Range of f

⬚

Range of f

g(x

)

g(x

)

f [g(x

)]

⬚

cob19545_ch03_353-370.qxd 8/10/10 7:49 PM Page 354

Most graphing calculators have the ability to evaluate a composition of functions at

a given point. From Example 2, enter f(x) as and g(x) as .

To evaluate the composition (since we know is in the domain), we can

(1) return to the home screen, enter and press (Figure 3.72), or

(2) enter on the screen and use the TABLE feature (Figure 3.73).

See Exercises 19 through 24.

⫽ Y

1X22

ENTER

1922

x ⫽ 91f ⴰ g2192

⫽ 3x ⫹ 2Y

⫽ 1x ⫺ 4

3–75 Section 3.6 The Composition of Functions and the Difference Quotient 355

College Algebra G&M—

Figure 3.72

Figure 3.73

EXAMPLE 3

䊳

Finding a Composition of Functions

For and , analyze the domain of

a. b. c. Find the actual compositions and comment.

Solution

䊳

a. : For g to be deﬁned, is our ﬁrst restriction. Once g(x) is used as

the input, we have , and additionally note that g(x) cannot

equal 1. This means , so . The domain of is all real numbers

except and .

b. : For f to be deﬁned, is our ﬁrst restriction. Once f(x) is used as

the input, we have , and additionally note that f(x) cannot be 0.

This means , so . The domain of is all real numbers

except and .

c. For :

composition of

with

substitute for

(

)

simplify denominator; invert and multiply

result

Notice the function rule for has an implied domain of , but does

not show that g (the inner function) is undeﬁned when (see Part a). The

domain of is actually all real numbers except and .x ⫽ 2x ⫽ 01

f ⴰ g21x2

x ⫽ 0

x ⫽ 21

f ⴰ g21x2

⫽

2 ⫺ x

⫽

2 ⫺ x

⫽

2 ⫺ x

⫽

b a

b⫺ 1

f 3g1x24⫽

3g1x2

g1x2⫺ 1

1f ⴰ g21x2

x ⫽ 1x ⫽ 0

1g ⴰ f

21x2x ⫽ 0

x ⫺ 1

⫽ 0

g3f

1x24⫽

f 1x2

x ⫽ 11g ⴰ f 21x2

x ⫽ 2x ⫽ 0

f ⴰ gx ⫽ 2

⫽ 1

f 3g1x24⫽

3g1x2

g1x2⫺ 1

x ⫽ 01

f ⴰ g21x2

1g ⴰ f 21x21f ⴰ g21x2

g1x2⫽

1x2⫽

x ⫺ 1

cob19545_ch03_353-370.qxd 8/10/10 7:50 PM Page 355

For we have:

composition of

with

substitute for

(

)

invert and multiply

result

Similarly, the function rule for has an implied domain of , but

does not show that f (the inner function) is undeﬁned when (see Part b).

The domain of is actually all real numbers except and .

Now try Exercises 25 through 30

䊳

As Example 3 illustrates, the domain of cannot simply be taken from

the new function rule for h. It must be determined from the functions composed to

obtain h. The graph of is shown in Figure 3.74. We can easily see is not

in the domain as there a vertical asymptote at . The fact that is also

excluded is obscured by the y-axis, but the table shown in Figure 3.75 conﬁrms that

is likewise not in the domain.x ⫽ 0

x ⫽ 0x ⫽ 2

x ⫽ 21

f ⴰ g21x2

h1x2⫽ 1

f ⴰ g21x2

x ⫽ 1x ⫽ 01g ⴰ f 21x2

x ⫽ 1

x ⫽ 01g ⴰ f 21x2

⫽

21x ⫺ 12

⫽

x ⫺ 1

⫽

x ⫺ 1

g3f

1x24⫽

f 1x2

1g ⴰ f

21x2

356 CHAPTER 3 Quadratic Functions and Operations on Functions 3–76

College Algebra G&M—

⫺10

Figure 3.74

Figure 3.75

To further explore concepts related to the domain of a composition, see Exer-

cises 69 and 70.

Decomposing a Composite Function

Based on Figure 3.76, would you say that the circle is inside the

square or the square is inside the circle? The decomposition of a

composite function is related to a similar question, as we ask

ourselves what function (of the composition) is on the “inside”—

the input quantity—and what function is on the “outside.” For

instance, consider where we see that is

“inside” the radical. Letting and we have

or f[g(x)].h1x2⫽ 1f ⴰ g21x2

1x2⫽ 1x,g1x2⫽ x ⫺ 4

x ⫺ 4h1x2⫽ 1x ⫺ 4

Figure 3.76

WORTHY OF NOTE

The decomposition of a function is

not unique and can often be done

in many different ways.

cob19545_ch03_353-370.qxd 8/10/10 7:50 PM Page 356

3–77 Section 3.6 The Composition of Functions and the Difference Quotient 357

College Algebra G&M—

EXAMPLE 4

䊳

Decomposing a Composite Function

Given identify two functions f and g so that

then check by composing the functions to obtain h(x).

Solution

䊳

Noting that is inside the squaring function, we assign g(x) as this inner

function: The outer function is the squaring function decreased

by 3, so

Check:

(

) is an input for

squares inputs, then decreases the result by 3

substitute for

(

)

✓

Now try Exercises 31 through 34

䊳

B. A Numerical and Graphical View

of the Composition of Functions

Just as with the sum, difference, product, and quotient of functions, the composition of

functions can also be interpreted and understood graphically. For , once the

value of g(x) is known (read from the graph), the value of , can also

be determined.

EXAMPLE 5

䊳

Interpreting the Composition of Functions Numerically

For f(x) and g(x) as shown, use the graph given to

determine the value of each expression.

a. f(4), g(2), and (2)

b. g(6), f(8), and (8)

c. (8) and (0)

d. (9) and (0)

Solution

䊳

a. For f(4), we go to and note that is a

point on the red graph: . For g(2), go to and note that (2, 4)

is a point on the blue graph: Since and ,

b. For and we ﬁnd that is a point on the blue graph: .

For and note that that (8, 6) is a point on the red graph: . Since

and , : .

c. As illustrated in parts (a) and (b), for we ﬁrst determine

g(8), then substitute this value into f. From the blue graph , and

. For we ﬁrst determine f(0), then substitute

this value into g. From the red graph , and .

d. To evaluate we follow the same sequence as before. From

the graph of f, and , showing . The same ideas

are applied to . Since and ,

Now try Exercises 35 and 36

䊳

As with other operations on functions, a composition can be understood graphi-

cally at speciﬁc points, as in Example 5, or for all allowable x-values. Exploring the

second option will help us understand certain transformations of functions, and why

1g ⴰ g2102⫽⫺2

g142⫽⫺2g102⫽ 41g ⴰ g2102⫽ g3g1024

1f ⴰ f

2192⫽ 1f 152⫽ 1f 192⫽ 5

1f ⴰ f

2192⫽ f 3f 1924

g1⫺12⫽ 1f

102⫽⫺1

1g ⴰ f

2102⫽ g3f 1024f 1⫺22⫽ 3

g182⫽⫺2

1f ⴰ g2182⫽ f

3g1824

1g ⴰ f

2182⫽⫺41g ⴰ f 2182⫽ g3f 1824⫽ g162⫽⫺4g162⫽⫺4f 182⫽ 6

182⫽ 6f 182, x ⫽ 8

g162⫽⫺416, ⫺42g162, x ⫽ 6

⫽ f

142⫽⫺1: 1f ⴰ g2122⫽⫺11f ⴰ g2122⫽ f 3g1224

142⫽⫺1g122⫽ 4g122⫽ 4.

x ⫽ 2f

142⫽⫺1

14, ⫺12x ⫽ 4

1g ⴰ g21f ⴰ f

1g ⴰ f 21f ⴰ g2

1g ⴰ f 2

1f ⴰ g2

f ⴰ g21x2⫽ f 3g1x24

f ⴰ g21x2

⫽ h1x2

x ⫹ 1 ⫽ 31

x ⫹ 14

⫺ 3

⫽ 3g1x24

⫺ 3

1f ⴰ g21x2⫽ f 3g1x24

1x2⫽ x

⫺ 3.

g1x2⫽ 1

x ⫹ 1.

x ⫹ 1

f ⴰ g21x2⫽ h1x2,

h1x2⫽ 11

x ⫹ 12

⫺ 3,

A. You’ve just seen how we

can compose two functions

and determine the domain, and

decompose a function

10⫺2

⫺4

g(x)

f(x)

cob19545_ch03_353-370.qxd 11/25/10 3:46 PM Page 357

the domain of a composition is deﬁned so carefully. For instance, consider the functions

and . The graph of f is a parabola opening downward

with vertex (0, 3). For we have , which

we recognize as the graph of f shifted 2 units to the right. Using ,

, and , we can deactivate Y

so the only the graphs of Y

andY

(in bold) are shown (see Figure 3.77). Here the domain of the composition was

not a concern, as both f and g have domain . But now consider ,

, and . The domain of g must exclude (which is

also excluded from the domain of h), while the domain of f is . But after

assigning , , and , the graph of Y

shows

a noticeable gap between and (Figure 3.78). The reason is that

uses g(x) as the input for f, meaning for the domain,

becomes . Solving this inequality graphically (Figure 3.79)

shows only for the intervals , leaving the gap seen

in Y

for the interval (2, 5]. The domain of h is .x 僆 1⫺q, 22 ´ 35, q2

x 僆 1⫺q, 22 ´ 35, q2g1x2ⱖ⫺1

g1x2ⱖ⫺1 S

2 ⫺ x

ⱖ⫺1

x ⱖ⫺11f ⴰ g21x2⫽ f

3g1x24

x ⫽ 5x ⫽ 2

⫽ Y

1X22Y

⫽

2 ⫺ X

⫽ 42X ⫹ 1

x ⱖ⫺1

x ⫽ 2h1x2⫽ 1f ⴰ g21x2g1x2⫽

2 ⫺ x

1x2⫽ 41x ⫹ 1x 僆⺢

⫽ Y

1X22Y

⫽ X ⫺ 2

⫽⫺X

⫹ 3

h1x2⫽ ⫺1x ⫺ 22

⫹ 3h1x2⫽ 1f ⴰ g21x2

g1x2⫽ 1x ⫺ 22f

1x2⫽ ⫺x

⫹ 3

358 CHAPTER 3 Quadratic Functions and Operations on Functions 3–78

College Algebra G&M—

⫺5

Figure 3.77

⫺5

Figure 3.78

⫺5

Figure 3.79

EXAMPLE 6

䊳

Interpreting a Composition Graphically and Understanding the Domain

Given , ,

a. State the domains of f and g.

b. Use a graphing calculator to study the graph of .

c. Algebraically determine the domain of h, and reconcile it with the graph.

Solution

䊳

a. The domain of g must exclude , and for the domain of f we must have

b. For the graph of h, enter , , and .

Graphing Y

on the 6:ZStandard screen shows a gap between

and (Figure 3.80).

c. Since the composition uses g(x) as the input for f,

(the domain for f ) becomes . Figure 3.81

shows a graphical solution to this inequality, which indicates for

. The domain of h is .x 僆 1⫺q, ⫺32 ´ 31, q2x 僆 1⫺q, ⫺32 ´ 31, q2

g1x2ⱕ 1

g1x2ⱕ 1 S

x ⫹ 3

ⱕ 1x ⱕ 1

1f ⴰ g21x2⫽ f

3g1x24

x ⫽ 1

x ⫽⫺3

ZOOM

⫽ Y

1X22Y

⫽

x ⫹ 3

⫽ 321 ⫺ x

x ⱕ 1

x ⫽⫺3

h1x2⫽ 1

f ⴰ g21x2

g1x2⫽

x ⫹ 3

f 1x2⫽ 311 ⫺ x

cob19545_ch03_353-370.qxd 11/25/10 3:47 PM Page 358

3–79 Section 3.6 The Composition of Functions and the Difference Quotient 359

College Algebra G&M—

⫺10

Figure 3.80

⫺10

Figure 3.81

Now try Exercises 37 and 38

䊳

C. Average Rates of Change and the Difference Quotient

We now return to our discussion of average rates of change, seeking to make the cal-

culation more efﬁcient due to (1) the large number of times the concept is applied, and

(2) its value to virtually all ﬁelds of study. Our current calculation

works very well, but requires us to recalculate for each new interval chosen. Using

a slightly different approach, we can develop a formula for a given function, and use it

to calculate various rates of change with greater efﬁciency. This is done by selecting a

point in the domain, and a point , where h is assumed to be some

very small, arbitrary number. Making these substitutions in our current formula gives

average rate of change

substitute for

for

simplify

The advantage of this new formula, called the difference quotient, is that the result is

a new function that can be evaluated repeatedly for any interval.

The Difference Quotient

For a given function f(x) and constant ,

is the difference quotient for f.

Note that the formula has three parts: (1) the function evaluated at :

(2) the function f itself: f(x), and (3) the constant h. The expression for is

actually a composition of functions, which can be evaluated and simpliﬁed prior to its

use in the difference quotient.

1x ⫹ h2⫺ f 1x2

1x ⫹ h2

1x ⫹ h2,x ⫹ h

1x ⫹ h2⫺ f 1x2

h ⫽ 0

⫽

1x ⫹ h2⫺ f 1x2

x ⫹ h ⫽

1x ⫹ h2⫺ f 1x2

1x ⫹ h2⫺ x

¢y

¢x

⫽

2⫺ f 1x

⫺ x

⫽ x ⫹ hx ⫽ x

¢y

¢x

¢y

¢x

⫽

2⫺ f 1x

⫺ x

B. You’ve just seen how

we can interpret the composi-

tion of functions numerically

and graphically

(1)

(2)

(3)

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