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5 Function Notation: The Concept of Function as a Rule

# 5 Function Notation: The Concept of Function as a Rule

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SECTION 1.5

Function Notation: The Concept of Function as a Rule

53

Function Notation

A function f is a rule that assigns to each input exactly one output. If we

write x for the input and y for the output, then we use the following notation

to describe f :

Function Input Output

T

T T

f 1x2 = y

The symbol f 1x2 is read “f of x” or “f at x” and is called the value of f at x or the image of x under f. This notation emphasizes the dependence of the output on the corresponding input—namely, the output f 1x2 is the result of applying the rule f to the

input x:

x S f 1x2

The following examples illustrate the meaning of function notation.

e x a m p l e 1 Function Notation

Consider the function f 1x2 = 8x.

(a) What is the name of the function?

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

(c) What rule does this function represent?

(d) Find f 122 . What does f 122 represent?

Solution

(a)

(b)

(c)

(d)

The name of the function is f.

The input is x, and the output is 8x.

The rule is “Multiply the input by 8.”

f 12 2 = 8 ؒ 2 = 16. So 16 is the value of the function at 2.

NOW TRY EXERCISE 9

e x a m p l e 2 Writing Function Notation

Express the given rule in function notation.

(a) “Multiply by 2, then add 5.”

(b) “Add 3, then square.”

Solution

(a) First we need to choose a letter to represent this rule. So let g stand for the rule

“Multiply by 2, then add 5.” Then for any input x, multiplying by 2 gives 2x,

then adding 5 gives 2x + 5. Thus we can write

g1x2 = 2x + 5

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

2x + 5.

54

CHAPTER 1

Data, Functions, and Models

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

any input x, adding 3 gives x + 3, then squaring gives 1x + 32 2. Thus we

have

h 1x2 = 1x + 32 2

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

1x + 32 2.

NOW TRY EXERCISE 13

e x a m p l e 3 Expressing a Function in Words

Give a verbal description of the given function.

(a) f 1x2 = x - 5

(b) g1x2 =

x +3

2

Solution

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

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

NOW TRY EXERCISE 17

e x a m p l e 4 Dependent and Independent Variables

(a) Express the equation y = x 2 + 2x in function notation, where x is the independent variable and y is the dependent variable.

(b) Express the function f 1x2 = 5x + 1 as an equation in two variables. Identify

the dependent and independent variables.

Solution

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.

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

x 2 + 2x. If we call this rule g, then we can write this function as

g1x2 = x 2 + 2x

(b) If we write y for the output f 1x 2 , then we can write the function as

y = 5x + 1

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

2

NOW TRY EXERCISES 23 AND 29

■ Evaluating Functions—Net Change

In function notation the input x plays the role of a placeholder. For example, the function f 1x2 = 3x 2 + x can be thought of as

f 1 ٗ 2 = 31 ٗ 2 2 +

ٗ

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:

f 1x 2 = 3x 2 + x

f 1z 2 = 3z 2 + z

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

e x a m p l e 5 Evaluating a Function

Let f 1x2 = 3x 2 + x. Evaluate the function at the given input.

(a) f 1- 22

(b) f 10 2

(c) f 14 2

(d) f A 12 B

Solution

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

(a)

(b)

(c)

(d)

f 1- 22 = 3 # 1- 2 2 2 + 1- 22 = 10

f 10 2 = 3 # 102 2 + 0 = 0

f 14 2 = 3 # 142 2 + 4 = 52

f A 12 B = 3 # A 12 B 2 + 12 = 54

NOW TRY EXERCISE 39

Function notation is a useful and practical way of describing the rule of a function; 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

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

a … b) is given by

f 1b2 - f 1a2

In the next example we find the net change in the weight of an astronaut at different elevations above the earth.

e x a m p l e 6 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

w 1h2 = 130 a

2

3960

b

3960 + h

(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?

(c) Find the net change in the weight of the astronaut from an elevation of

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

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

w 1h2 = 130 a

w 11002 = 130 a

2

3960

b

3960 + h

Function

2

3960

b

3960 + 100

Replace h by 100

L 123.67

h

w(h)

0

100

200

300

400

500

130

124

118

112

107

102

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.

(c) The net change in the weight of the astronaut between these two elevations is

w14002 - w11002 . We use the entries already calculated in the table in part (b)

to evaluate this net change.

w14002 - w11002 L 107 - 124

From the table

= - 17

The net change is - 17 lb. The negative sign means that the astronaut’s weight

decreased by about 17 lb.

2

NOW TRY EXERCISES 35 AND 61

■ 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

f 1x2 = x 2

0…x…5

then the domain is the set of real numbers x for which 0 … x … 5. 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

f 1x2 =

1

x -4

g1x 2 = 1x

The function f is not defined when x is 4, so its domain is 5x Έ x

is not defined for negative x, so its domain is 5x Έ x Ú 06.

e x a m p l e 7 Finding the Domain of a Function

Find the domain of each function.

(a) f 1x2 =

1

x1x - 22

(b) g1x2 = 1x - 1

46. The function g

SECTION 1.5

Function Notation: The Concept of Function as a Rule

57

Solution

Solving inequalities is reviewed

in Algebra Toolkit C.3, page T62.

(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 5x Έ x 0 and x 26.

(b) The function is defined only when x - 1 Ú 0. This means that x Ú 1. So the

domain of g is 5x Έ x Ú 16.

2

NOW TRY EXERCISES 49 AND 51

■ Piecewise Defined Functions

A piecewise defined function is a function that is defined by different rules on different 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 8

Cell Phone Plan

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

C 1x2 = e

39

39 + 0.201x - 4002

if 0 … x … 400

if x 7 400

Find (a) C 11002 , (b) C 14002 , and (c) C 14802 .

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 0 … x … 400, then the value of C(x) is 39.

On the other hand, if x 7 400, the value of C(x) is 39 + 0.201x - 4002 .

(a) Since 100 … 400, we have C11002 = 39.

(b) Since 400 … 400, we have C14002 = 39.

(c) Since 480 7 400, we have C14802 = 39 + 0.201480 - 4002 = 55. Thus, the

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

minutes.

Aaron Kohr/Shutterstock.com 2009

IN CONTEXT ➤

The California aqueduct

NOW TRY EXERCISES 55 AND 67

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 distant 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, including 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.

58

CHAPTER 1

Data, Functions, and Models

e x a m p l e 9 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 x 6 4000 and for x Ú 4000. For x gallons

the cost is 0.008x if x 6 4000 and 0.012x if x Ú 4000. So we can express the

function C as

C1x 2 = e

0.008x

0.012x

if x 6 4000

if x Ú 4000

(b) Since 3900 6 4000, we have C139002 = 0.008139002 = 31.20. Since

4200 7 4000, we have C142002 = 0.012142002 = 50.40. So using 3900

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

NOW TRY EXERCISE 69

1.5 Exercises

CONCEPTS

Fundamentals

1. If a function f is given by the equation y = f 1x 2 , then the independent variable is

_______, the dependent variable is _______, and f 1a2 is the _______ of f at a.

2. If f 1x2 = x 2 + 1, then f 122 = _______ and f 102 = _______.

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

So if f 1x2 = x 2 + 1, then the net change in the value of the function f when x changes

from 0 to 2 is the difference ____ Ϫ ____ ϭ ____.

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?

f 1x 2 = x 2 - 3x

g 1x 2 =

x -5

x

h 1x 2 = 1x - 10

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

function at 5.

x

f (x)

0

19

2

4

6

6. A function is given algebraically by the formula f 1x 2 = 1x - 4 2 2 + 3. Fill in the table

in the margin to give a numerical representation of f.

7. How would you describe the quantity f 1b 2 - f 1a 2 without using function notation? For

example, how would you describe the net change between - 2 and 5 for the function

y = x 2 - 2 without function notation?

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.

SKILLS

9–12 ■

(a)

(b)

(c)

(d)

A function is given.

What is the name of the function?

What letter represents the input? What is the output?

What rule does this function represent?

Find f 1102 . What does f 1102 represent?

9. f 1x2 = 2x + 1

13–16

10. g1w 2 = w 3 - 1

11. h1z 2 = 1z - 2 2 2

12. A1r2 =

r2

-7

5

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. f 1x2 =

18. g1x 2 =

x

+7

2

x+7

2

19. h1w2 = 7w 2 - 5

20. k1w2 = 1w + 12 2 - 9

21. A1r 2 = 51r + 32 3

22. V 1r 2 =

2

+1

r

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. y = 5x

24. y = 3x 3

25. x = 4y 3

26. x = y 3 - 1

27. y = 3x 2

28. x = 4y 2

29–34

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

and dependent variables.

29. f 1x2 = 3x 2 - 1

31. f 1w2 = 3w 2 - 1

30. g1x 2 = 2x 3 - 2x

32. g1z 2 = 2z 3 - 2z

33. S1r 2 = 4pr 2

34. V 1r2 = 43pr 3

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. f 1x2 = 2x 2 - 7

x

f(x)

-2

-1

36. g1s 2 = 3s - 5

0

1

2

s

g(s)

-4

-2

0

1

2

60

CHAPTER 1

Data, Functions, and Models

37. h1z 2 = 1z + 32 3 - 2

z

-3

-1

38. k1t2 =

0

1

t

2

h(z)

39–42

1

t +1

0

1

2

3

4

k(t)

Evaluate the function at the indicated values.

39. f 1x2 = x 2 + 2x:

3

(a) f 10 2

2

40. g1x2 = x - 4x :

(a) g12 2

(b) f 1 12 2

(c) f 1- 22

(d) f 1a 2

(b)

(c) g1- 12

(d) g1b2

gA 12 B

41. r1x2 = 12x - 1:

(a) r12 2

(b) r15 2

(c) r13 2

(d) r1c 2

42. s1x 2 =

(a) s10 2

(b) s13 2

(c) s14 2

(d) s1a 2

1

:

1x + 1

43. The surface area S of a sphere is a function of its radius r given by S1r2 = 4pr 2.

(a) Find S122 and S132 .

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

(c) Find the net change in the value of the function S when r changes from 1 to 4.

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

given by V 1r 2 = 2pr 3.

(a) Find V 142 and V 192 .

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

(c) Find the net change in the value of the function V when r changes from 1 to 4.

45–54

Find the domain of the function.

45. f 1x2 = 2x

46. f 1x2 = x 2 + 1

47. g1x2 = 2x, - 1 … x … 5

49. h1x2 =

1

x-3

48. g1x2 = x 2 + 1, 0 … x … 5

50. h1x2 =

1

3x - 6

51. k1x 2 = 1x - 5

52. k1x 2 = 1x + 9

53. r1x 2 =

54. r1x 2 =

55–60

3

1x - 4

12 + x

3 -x

Evaluate the piecewise defined function at the indicated values.

55. f 1x2 = e

if x 6 0

if x Ú 0

(b) f 10 2

(c) f 1- 22

(d) f 11 2

56. g1x2 = e

if x 6 0

if x Ú 0

(b) g10 2

(c) g1- 2 2

(d) g11 2

(c) h13 2

(d) h11 2

(c) k1- 32

(d) k17 2

-x

x

(a) f 1- 1 2

- 2x

2x

(a) g1- 12

57. h1x2 = e

2

x

x+1

(a) h1- 3 2

58. k1x 2 = e

(a) k122

5

2x - 3

if x 6 1

if x Ú 1

(b) h10 2

if x … 2

if x 7 2

(b) k10 2

SECTION 1.5

59. F 1x2 = e

(a) F 12 2

x2 + 2x

2x

Function Notation: The Concept of Function as a Rule

if x … - 1

if x 7 - 1

(b) F 1- 12

3x

if x 6 0

60. G 1x2 = c x + 1

if 0 … x … 2

2

if x 7 2

1x - 22

(a) G 1- 2 2

(b) G 1- 12

CONTEXTS

(c) F 10 2

(d) F 1- 22

(c) G 10 2

(d) G 13 2

61

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

D 1h2 = 27920h + h 2

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.)

(c) Commercial aircraft fly at an altitude of about 7 mile. How far can the pilot see?

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

V 1t 2 = 50 a 1 -

t 2

b

20

0 … t … 20

(a) Find V102 and V1202 . What do these values represent?

(b) Make a table of values of V1t2 for t ϭ 0, 5, 10, 15, 20.

(c) Find the net change of the volume of water in the tank when t changes from 0 to 10.

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

h 1t2 = 13,000 - 16t 2

h(t)=13,000-16t™

(a) Find h 1102 and h 1202 . 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?

(c) Find the net change in the sky diver’s height from 0 to 25 seconds.

62

CHAPTER 1

Data, Functions, and Models

64. Path of a Ball

function

A baseball is thrown across a playing field. Its path is given by the

h 1x2 = - 0.005x 2 + x + 5

where x is the distance the ball has traveled horizontally and h 1x 2 is its height above

ground level, both measured in feet.

(a) Find h 1102 and h 11002 . What do these values represent?

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

(c) Find the net change in the height of the ball as the horizontal distance x changes

from 10 to 100 ft.

x

(year)

R(x)

(billions of dollars)

2000

2001

2002

2003

2004

2005

2006

7.66

8.41

9.52

9.49

9.54

8.99

9.49

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 R 120002 and R 120062 . What do these values represent?

(b) For what values of x is R 1x2 = 9.49?

(c) Find the net change in box office revenues from 2000 to 2006.

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 f 119952 , f 120052 , g11995 2 , and g120052 . What do these values represent?

(b) Find x and y so that f 1x2 = f 1y 2 .

(c) Find the net change in biotechnology revenues and in R&D expenses from 1995

to 2005.

x

(year)

f(x)

(billions of dollars)

g(x)

(billions of dollars)

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

12.7

14.6

17.4

20.2

22.3

26.7

29.6

29.6

39.2

43.8

50.7

7.7

7.9

9.0

10.6

10.7

14.2

15.7

20.5

17.9

19.6

19.8

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

function of income x (in dollars):

0

T 1x2 = • 0.08x

1600 + 0.151x - 20,0002

if 0 … x … 10,000

if 10,000 6 x … 20,000

if 20,000 6 x

(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.

(c) If a businessman pays \$25,000 in taxes in this country, what is his income?

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

- 13

0.025

x

i 1x2 =

e 12

0.0368

x

12

if 0 … x 6 2500

if 2500 … x … 20,000

if 20,000 6 x

(a) Find i 110,0002 and i 120,0002 . What do these values represent?

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

(c) Find the balance Greg needs to have in his account to earn \$70 interest in one month.

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:

C1x 2 = e

___________

___________

if x 6 100

if x Ú 100

(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:

T1x2 = e

___________

___________

if 0 … x … 2

if 2 6 x

(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:

_________

F1x2 = c _________

_________

if 0 6 x 6 40

if 40 … x … 65

if 65 6 x

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

(c) A driver is assessed a fine of \$225. What are the two possible speeds at which he

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:

E1x2 = e

___________

___________

if 0 … x … 350

if 350 6 x

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

(c) One bill for electric usage is \$65.67. How many kilowatt hours are covered by this bill?

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