Stewart J. College Algebra: Concepts and Contexts

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54 CHAPTER 1

■

Data, Functions, and Models

example

Expressing a Function in Words

Give a verbal description of the given function.

(a)

(b)

Solution

(a) The rule f is “Subtract 5 from the input.”

(b) The rule is “Add 3 to the input, then divide by 2.”

■ NOW TRY EXERCISE 17 ■

g1x 2=

x + 3

f 1x2= x - 5

example

Dependent and Independent Variables

■ Evaluating Functions—Net Change

In function notation the input x plays the role of a placeholder. For example, the func-

tion can be thought of as

ⵧ

2= 31

ⵧ

f 1x2= 3x

+ x

(a) Express the equation in function notation, where x is the inde-

pendent variable and y is the dependent variable.

(b) Express the function as an equation in two variables. Identify

the dependent and independent variables.

Solution

(a) The equation determines a function in which each input x gives the output

. If we call this rule , then we can write this function as

(b) If we write y for the output , then we can write the function as

The dependent variable is y, and the independent variable is x.

■ NOW TRY EXERCISES 23 AND 29 ■

y = 5x + 1

f 1x2

g1x 2= x

+ 2x

f 1x2= 5x + 1

y = x

+ 2x

Although we chose the letter x for

the independent variable, any letter

can be chosen. The function (or

rule) is the same regardless of the

letter we choose to describe it.

(b) Let’s choose the letter h to stand for the rule “Add 3, then square.” Then for

any input x, adding 3 gives , then squaring gives . Thus we

have

Note that the input is the number x and the corresponding output is the number

■ NOW TRY EXERCISE 13 ■

1x + 3 2

h 1x2= 1x + 32

1x + 3 2

x + 3

SECTION 1.5

■

Function Notation: The Concept of Function as a Rule 55

So any letter can be used for the input of a function. Both of the following are the

same function:

We can see that each of these represent the same rule applied to the input.

1z2= 3z

+ z

f 1x 2= 3x

+ x

example

Evaluating a Function

Let . Evaluate the function at the given input.

(a) (b) (c) (d)

Solution

To evaluate f at a number, we substitute the number for x in the definition of f.

(a)

(b)

(c)

(d)

■ NOW TRY EXERCISE 39 ■

Function notation is a useful and practical way of describing the rule of a func-

tion; we’ll see many examples of the advantages of function notation in the coming

chapters. Here we’ll see how function notation allows us to write a concise formula

for the net change in the value of a function between two points.

Net Change in the Value of a Function

f A

B= 3

f 14 2= 3

14 2

+ 4 = 52

f 10 2= 3

10 2

+ 0 = 0

f 1- 22= 3

1- 22

+ 1- 22= 10

f A

Bf 14 2f 102f 1- 22

f 1x 2= 3x

+ x

The net change in the value of a function f as x changes from a to b (where

) is given by

f 1b 2- f 1a 2

a … b

In the next example we find the net change in the weight of an astronaut at dif-

ferent elevations above the earth.

example

Net Change in the Weight of an Astronaut

If an astronaut weighs 130 pounds on the surface of the earth, then her weight when

she is h miles above the earth is given by the function

(a) Evaluate w(100). What does your answer mean?

(b) Construct a table of values for the function w that gives the astronaut’s weight

at heights from 0 to 500 miles. What do you conclude from the table?

100 miles to an elevation of 400 miles. Interpret your result.

1h 2= 130 a

3960

3960 + h

56 CHAPTER 1

■

Data, Functions, and Models

Solution

(a) We want the value of the function w when h is 100; substituting 100 for h in

the definition of w, we get

Function

Replace h by 100

Calculator

This means that at a height of 100 mi, she weighs about 124 lb.

(b) The table gives the astronaut’s weight, rounded to the nearest pound, at

100-mile increments. The values in the table in the margin are calculated as in

part (a). The table indicates that the higher the astronaut ascends, the less she

weighs.

. We use the entries already calculated in the table in part (b)

to evaluate this net change.

From the table

The net change is lb. The negative sign means that the astronaut’s weight

decreased by about 17 lb.

■ NOW TRY EXERCISES 35 AND 61 ■

- 17

=-17

w14002- w11002L 107 - 124

w1400 2- w1100 2

L 123.67

w 11002= 130 a

3960

3960 + 100

w 1h2= 130 a

3960

3960 + h

h w(h)

0 130

100 124

200 118

300 112

400 107

500 102

■ The Domain of a Function

The domain of a function is the set of all inputs for the function. The domain may

be stated explicitly. For example, if we write

then the domain is the set of real numbers x for which If the domain of

a function is not given explicitly, then by convention the domain is the set of all real

number inputs for which the output is defined. For example, consider the functions

The function f is not defined when x is 4, so its domain is . The function

is not defined for negative x, so its domain is .5x

冨 x Ú 06

g5x 冨 x  46

f 1x 2=

x - 4

g1x 2= 1x

0 … x … 5.

f 1x 2= x

0 … x … 5

example

Finding the Domain of a Function

Find the domain of each function.

(a) (b) g1x 2= 1x - 1

f 1x 2=

x1x - 22

SECTION 1.5

■

Function Notation: The Concept of Function as a Rule 57

Solving inequalities is reviewed

in Algebra Toolkit C.3, page T62.

Solution

(a) The function is not defined when the denominator is 0, that is, when x is 0 or

2. So the domain of f is and .

(b) The function is defined only when . This means that . So the

domain of is .

■ NOW TRY EXERCISES 49 AND 51 ■

5x 冨 x Ú 16g

x Ú 1x - 1 Ú 0

x ⫽ 265x 冨 x ⫽ 0

■ Piecewise Defined Functions

A piecewise defined function is a function that is defined by different rules on dif-

ferent parts of its domain. The rules for such functions cannot be expressed as a

single algebraic equation. This is one of the many reasons that we need function

notation.

example

Cell Phone Plan

A cell phone plan has a basic charge of $39 per month. The plan includes 400 min-

utes and charges 20 cents for each additional minute of usage. The monthly charges

are a function of the number x of minutes used, given by

Find (a) , (b) , and (c) .

Solution

Remember that a function is a rule. Here is how we apply the rule for this function.

First we look at the value of the input x. If , then the value of C(x) is 39.

On the other hand, if , the value of C(x) is .

(a) Since , we have .

(b) Since , we have .

plan charges $39 for 100 minutes, $39 for 400 minutes, and $55 for 480

minutes.

■ NOW TRY EXERCISES 55 AND 67 ■

C1480 2= 39 + 0.201480 - 4002= 55480 7 400

C1400 2= 39400 … 400

C1100 2= 39100 … 400

39 + 0.201x - 4002x 7 400

0 … x … 400

C 1480 2C 1400 2C 11002

C 1x 2= e

39 if 0 … x … 400

39 + 0.201x - 4002 if x 7 400

Some of the fastest-growing cities in the United States are located in areas where

water is in short supply. In the Southwest, water is brought to many cities from dis-

tant rivers and lakes via aqueducts. City planners must regulate growth carefully to

ensure that adequate water sources exist for new developments. On average, each

U.S. resident uses between 40 and 80 gallons of water daily. In arid areas people tend

to use less water. In Arizona, for example, people conserve water in many ways, in-

cluding the use of xeriscaping (landscaping with desert-friendly plants).

To discourage excessive water use in arid areas, cities charge different rates that

depend on the amount of water used. In the next example we model the domestic cost

of water using a piecewise defined function.

IN CONTEXT ➤

The California aqueduct

Aaron Kohr/Shutterstock.com 2009

58 CHAPTER 1

■

Data, Functions, and Models

example

Water Rates

To discourage excessive water use, a city charges its residents $0.008 per gallon for

households that use less than 4000 gallons a month and $0.012 for households that

use 4000 gallons or more a month.

(a) Find a piecewise defined function C that gives the water bill for a household

using x gallons per month.

(b) Find C(3900) and C(4200). What do your answers represent?

Solution

(a) Since the cost of x gallons of water depends on the usage, we need to define

the function C in two pieces: for and for . For x gallons

the cost is 0.008x if and 0.012x if . So we can express the

function C as

(b) Since , we have . Since

, we have . So using 3900

gallons costs $31.20; using 4200 gallons costs $50.40.

■ NOW TRY EXERCISE 69 ■

C14200 2= 0.01214200 2= 50.404200 7 4000

C13900 2= 0.00813900 2= 31.203900 6 4000

C1x 2= e

0.008x if x 6 4000

0.012x if x Ú 4000

x Ú 4000x 6 4000

1.5 Exercises

CONCEPTS

Fundamentals

1. If a function f is given by the equation , then the independent variable is

_______, the dependent variable is _______, and is the _______ of f at a.

2. If , then

_______ and _______.

3. The net change in the value of the function f from a to b is the difference

____  ____.

So if , then the net change in the value of the function f when x changes

from 0 to 2 is the difference

____  ____  ____.

4. For a function f, the set of all possible inputs is called the

_______ of f, and the set of

all possible outputs is called the

_______ of f.

5. (a) Which of the following functions have 5 in their domain?

(b) For the functions from part (a) that do have 5 in their domain, find the value of the

function at 5.

6. A function is given algebraically by the formula . Fill in the table

in the margin to give a numerical representation of f.

Think About It

7. How would you describe the quantity without using function notation? For

example, how would you describe the net change between and 5 for the function

without function notation?y = x

- 2

f 1b 2- f 1a 2

f 1x 2= 1x - 42

+ 3

f 1x 2= x

- 3xg 1x2=

x - 5

h 1x 2= 1x - 10

f 1x 2= x

+ 1

f 102=f 122=f 1x 2= x

+ 1

f 1a 2

y = f 1x2

x 0 2 4 6

f (x) 19

SECTION 1.5

■

Function Notation: The Concept of Function as a Rule 59

8. True or false?

(a) If the net change of a function f from a to b is zero, then the function must be

constant between a and b.

(b) If the net change of a function f from a to b is positive, then the values of the

function must steadily increase from a to b.

9–12

■ A function is given.

(a) What is the name of the function?

(b) What letter represents the input? What is the output?

(d) Find . What does represent?f 1102f 1102

SKILLS

- 2 - 1

0 1 2

f(x)

- 4 - 2

0 1 2

g (s)

9. 10. 11. 12.

13–16

■ Express the given rule in function notation.

13. “Add 2, then multiply by 5.” 14. “Divide by 3, then add 2.”

15. “Square, add 7, then divide by 4.” 16. “Subtract 5, square, then subtract 1.”

17–22 ■ Give a verbal description of the function.

17. 18.

19. 20.

21. 22.

23–28

■ An equation is given.

(a) Does the equation define a function with x as the independent variable and y as the

dependent variable? If so, express the equation in function notation with x as the

independent variable.

(b) Does the equation define a function with y as the independent variable and x as the

dependent variable? If so, express the equation in function notation with y as the

independent variable.

23. 24. 25.

26. 27. 28.

29–34

■ A function is given. Express the function as an equation. Identify the independent

and dependent variables.

29. 30. 31.

32. 33. 34.

35–38

■ A function is given.

(a) Complete the table of values for the function.

(b) Find the net change in the value of the function when x changes from 0 to 2.

35. 36. g1s 2= 3s - 5f 1x 2= 2x

- 7

V1r 2=

S1r 2= 4pr

g1z2= 2z

- 2z

f 1w 2= 3w

- 1g1x 2= 2x

- 2xf 1x2= 3x

- 1

x = 4y

y = 3x

x = y

- 1

x = 4y

y = 3x

y = 5x

V1r 2=

+ 1A1r 2= 51r + 32

k1w 2= 1w + 12

- 9h1w 2= 7w

- 5

g1x 2=

x + 7

f 1x 2=

+ 7

A1r 2=

- 7h1z2= 1z - 22

g1w 2= w

- 1f 1x2= 2x + 1

60 CHAPTER 1

■

Data, Functions, and Models

37. 38. k1t 2=

t + 1

h1z2= 1z + 32

- 2

- 3 - 1

0 1 2

h(z)

t 0 1 2 3 4

k(t)

39–42 ■ Evaluate the function at the indicated values.

39. : (a) (b) (c) (d)

40. : (a) (b) (c) (d)

41. : (a) (b) (c) (d)

42. : (a) (b) (c) (d)

43. The surface area S of a sphere is a function of its radius r given by

(a) Find and .

(b) What do your answers in part (a) represent?

44. The volume V of a can that has height two times the radius is a function of its radius r

given by

(a) Find and .

(b) What do your answers in part (a) represent?

45–54

■ Find the domain of the function.

45. 46.

47. 48.

49. 50.

51. 52.

53. 54.

55–60

■ Evaluate the piecewise defined function at the indicated values.

55.

(a) (b) (c) (d)

56.

(a) (b) (c) (d)

57.

(a) (b) (c) (d)

58.

(a) (b) (c) (d) k172k1- 32k102k122

k1x 2= e

5 if x … 2

2x - 3 if x 7 2

h112h132h102h1- 32

h1x 2= e

if x 6 1

x + 1 if x Ú 1

g112g1- 22g102g1- 12

g1x 2= e

- 2x if x 6 0

2x if x Ú 0

f 112f 1- 22f 102f 1- 12

f 1x 2= e

- x if x 6 0

x if x Ú 0

r1x 2=

12 + x

3 - x

r1x 2=

1x - 4

k1x 2= 1x + 9k1x 2= 1x - 5

h1x 2=

3x - 6

h1x 2=

x - 3

g1x 2= x

+ 1,0 … x … 5g1x 2= 2x,- 1 … x … 5

f 1x 2= x

+ 1f 1x2= 2x

V19 2V14 2

V1r 2= 2pr

S132S122

S1r 2= 4pr

s1a 2s142s132s102s1x2=

1x + 1

r1c 2r132r152r122r1x 2= 12x - 1

g1b 2g1- 12gA

Bg122g1x 2= x

- 4x

f 1a 2f 1- 22f 1122f 102f 1x 2= x

+ 2x

SECTION 1.5

■

Function Notation: The Concept of Function as a Rule 61

59.

(a) (b) (c) (d)

60.

(a) (b) (c) (d)

61. How Far Can You See? Because of the curvature of the earth, the maximum distance

D that one can see from the top of a tall building or from an airplane at height h is

modeled by the function

where D and h are measured in miles.

(a) Find D(0.1). What does this value represent?

(b) How far can one see from the observation deck of Toronto’s CN Tower, 1135 ft

above the ground? (Remember that one mile is 5280 ft.)

(d) Find the net change of the distance D as one climbs the CN Tower from a height of

100 ft to a height of 1135 ft.

62. Torricelli’s Law A tank holds 50 gallons of water, which drains from a leak at the

bottom, causing the tank to empty in 20 minutes. The tank drains faster when it is

nearly full because the pressure on the leak is greater. Torricelli’s Law models the

volume (in gallons) of water remaining in the tank after t minutes as

(a) Find and . What do these values represent?

(b) Make a table of values of for t  0, 5, 10, 15, 20.

V1t 2

V1202V102

V1t2= 50 a1 -

0 … t … 20

D 1h2= 27920h + h

G 13 2G 10 2G 1- 12G 1- 22

G 1x 2= c

3x if x 6 0

x + 1 if 0 … x … 2

1x - 22

if x 7 2

F 1- 22F 102F 1- 12F 122

F 1x 2= e

+ 2x if x …-1

2x if x 7-1

CONTEXTS

63. A Falling Sky Diver When a sky diver jumps out of an airplane from a height of

13,000 ft, her height h (in feet) above the ground after t seconds is given by the function

(a) Find and . What do these values represent?

(b) For safety reasons a sky diver must open the parachute at a height of about 2500 ft

(or higher). A sky diver opens her parachute after 24 seconds. Did she open her

parachute at a safe height?

h 1202h 1102

h 1t2= 13,000 - 16t

h(t)=13,000-16t™

62 CHAPTER 1

■

Data, Functions, and Models

(year)

R(x)

(billions of dollars)

2000 7.66

2001 8.41

2002 9.52

2003 9.49

2004 9.54

2005 8.99

2006 9.49

64. Path of a Ball A baseball is thrown across a playing field. Its path is given by the

function

where x is the distance the ball has traveled horizontally and is its height above

ground level, both measured in feet.

(a) Find and . What do these values represent?

(b) What is the initial height of the ball?

from 10 to 100 ft.

65. Movie Theater Revenue Box office revenues for movie theaters have remained high

for the last decade, in spite of the recent popularity of DVD players and home movie

theaters. The function R represented by the table in the margin shows the annual U.S.

box office revenue (in billions).

(a) Find and . What do these values represent?

(b) For what values of x is ?

66. Biotechnology The biotechnology industry is responsible for hundreds of medical,

environmental, and technological advances; one such advance is DNA fingerprinting. In

the table below, the function f gives annual biotechnology revenue and the function g

gives annual research and development (R&D) revenue, in billions of dollars, between

1995 and 2005.

(a) Find , , , and . What do these values represent?

(b) Find x and y so that .

to 2005.

f 1x 2= f 1y 2

g120052g119952f 120052f 11995 2

R 1x 2= 9.49

R 12006 2R 120002

h 1100 2h 110 2

h 1x

h 1x2=-0.005x

+ x + 5

(year)

f(x)

(billions of dollars)

g(x)

(billions of dollars)

1995 12.7 7.7

1996 14.6 7.9

1997 17.4 9.0

1998 20.2 10.6

1999 22.3 10.7

2000 26.7 14.2

2001 29.6 15.7

2002 29.6 20.5

2003 39.2 17.9

2004 43.8 19.6

2005 50.7 19.8

67. Income Tax In a certain country, income tax T is assessed according to the following

function of income x (in dollars):

(a) Find T(5000), T(12,000), and T(25,000). What do these values represent?

(b) Find the tax on an income of $20,000.

T 1x2= •

if 0 … x … 10,000

0.08x if 10,000 6 x … 20,000

1600 + 0.151x - 20,0002

if 20,000 6 x

SECTION 1.5

■

Function Notation: The Concept of Function as a Rule 63

68. Interest Income Greg earned $5000 over the summer working on a construction site. He

deposits his earnings in a money market account that offers a graduated interest rate based

on the balance of the account. If the balance is between $2500 and $20,000, the account

earns 2.5% interest per annum; if the balance is above $20,000, the account earns 3.68%;

and if the balance falls below $2500, the bank pays no interest but charges a $13 fee per

month. The interest i earned in 1 month on a balance of x dollars is modeled by the function

(a) Find and . What do these values represent?

(b) Find the interest Greg earns in the first month (on his initial $5000 deposit).

69. Internet Purchases An Internet bookstore charges $15 shipping for orders under

$100 but provides free shipping for orders of $100 or more. The cost C of an order is a

function of the total price x of the books purchased.

(a) Express C as a piecewise defined function:

(b) Find C(75), C(100), and C(105). What do these values represent?

70. Cost of a Hotel Stay A hotel chain charges $75 each night for the first two nights and

$50 for each additional night’s stay. The total cost T is a function of the number of

nights x that a guest stays.

(a) Express T as a piecewise defined function:

(b) Find T(2), T(3), and T(5). What do these values represent?

71. Speeding Tickets In a certain state the maximum speed permitted on freeways is

65 mi/h, and the minimum is 40 mi/h. The fine for violating these limits is $15 for every

mile above the maximum or below the minimum speed. So the fine F is a function of

the driving speed x (mi/h) on the freeway.

(a) Express F as a piecewise defined function:

(b) Find F(30), F(50), and F(75). What do these values represent?

could have been driving when he was caught?

72. Utility Charges A utility company charges a base rate of 10 cents per kilowatt hour

(kWh) for the first 350 kWh and 15 cents per kilowatt hour for all additional electricity

usage. The amount E that the utility company charges is a function of the number x of

kilowatt hours used.

(a) Express E as a piecewise defined function:

(b) Find E(300), E(350), and E(600). What do these values represent?

E1x 2= e

___________ if 0 … x … 350

___________ if 350 6 x

F1x 2= c

_________ if 0 6 x 6 40

_________ if 40 … x … 65

_________ if 65 6 x

T1x 2= e

___________ if 0 … x … 2

___________ if 2 6 x

C1x 2= e

___________ if x 6 100

___________ if x Ú 100

i 120,000 2i 110,0002

i 1x2=

- 13 if 0 … x 6 2500

0.025

x if 2500 … x … 20,000

0.0368

x if 20,000 6 x