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E. Applications of Inverse Functions

# E. Applications of Inverse Functions

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CHAPTER 5 Exponential and Logarithmic Functions

5.1 EXERCISES

CONCEPTS AND VOCABULARY

Fill in each blank with the appropriate word or phrase. Carefully reread the section if needed.

1. A function is one-to-one if each

coordinate

corresponds to exactly

first coordinate.

2. If every

line intersects the graph of a

function in at most

point, the function is

one-to-one.

3. A certain function is defined by the ordered pairs

1Ϫ2, Ϫ112, 10, Ϫ52, 12, 12 , and (4, 19). The inverse

function is

.

4. To find f Ϫ1 using the algebraic method, we (1) use

x and y,

(3)

for y and replace y with f Ϫ1 1x2 .

5. State true or false and explain why: To show that g

is the inverse function for f, simply show that

1 f ‫ ؠ‬g21x2 ϭ x. Include an example in your

response.

6. Discuss/Explain why no inverse function exists for

f 1x2 ϭ 1x ϩ 32 2 and g1x2 ϭ 24 Ϫ x2. How would

the domain of each function have to be restricted to

allow for an inverse function?

Determine whether each graph given is the graph of a

one-to-one function. If not, give examples of how the

definition of one-to-oneness is violated.

7.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

9.

5

4

3

2

1

11.

y

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

10.

1 2 3 4 5 x

12.

1 2 3 4 5 x

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

15. 51Ϫ7, 42, 1Ϫ1, 92, 10, 52, 1Ϫ2, 12, 15, Ϫ526

16. 519, 12, 1Ϫ2, 72, 17, 42, 13, 92, 12, 726

17. 51Ϫ6, 12, 14, Ϫ92, 10, 112, 1Ϫ2, 72, 1Ϫ4, 52, 18, 126

1 2 3 4 5 x

y

5

4

3

2

1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

14.

Determine whether the functions given are one-to-one.

If not, state why.

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

y

5

4

3

2

1

1 2 3 4 5 x

y

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

5

4

3

2

1

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

y

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

8.

13.

1 2 3 4 5 x

18. 51Ϫ6, 22, 1Ϫ3, 72, 18, 02, 112, Ϫ12, 12, Ϫ32, 11, 326

Determine if the functions given are one-to-one by noting

the function family to which each belongs and mentally

picturing the shape of the graph. If a function is not one-toone, discuss how the definition of one-to-oneness is violated.

19. f 1x2 ϭ 3x Ϫ 5

20. g1x2 ϭ 1x ϩ 22 3 Ϫ 1

23. s1t2 ϭ 12t Ϫ 1 ϩ 5

3

24. r1t2 ϭ 2

tϩ1Ϫ2

25. y ϭ 3

26. y ϭ Ϫ2x

21. h1x2 ϭ ϪͿx Ϫ 4Ϳ ϩ 3

22. p1t2 ϭ 3t2 ϩ 5

For Exercises 27 to 30, find the inverse function of the

one-to-one functions given.

27. f 1x2 ϭ 51Ϫ2, 12, 1Ϫ1, 42, 10, 52, 12, 92, 15, 1526

28. g1x2 ϭ 51Ϫ2, 302, 1Ϫ1, 112, 10, 42, 11, 32, 12, 226

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Section 5.1 One-to-One and Inverse Functions

29. v(x) is defined by the ordered pairs shown.

For each function f(x) given, prove (using a composition)

that g(x) ϭ f Ϫ1(x).

53. f 1x2 ϭ Ϫ2x ϩ 5, g1x2 ϭ

54. f 1x2 ϭ 3x Ϫ 4, g1x2 ϭ

30. w(x) is defined by the ordered pairs shown.

xϪ5

Ϫ2

xϩ4

3

3

55. f 1x2 ϭ 2

x ϩ 5, g1x2 ϭ x3 Ϫ 5

3

56. f 1x2 ϭ 2

x Ϫ 4, g1x2 ϭ x3 ϩ 4

57. f 1x2 ϭ 23x Ϫ 6, g1x2 ϭ 32x ϩ 9

58. f 1x2 ϭ 45x ϩ 6, g1x2 ϭ 54x Ϫ

15

2

59. f 1x2 ϭ x2 Ϫ 3; x Ն 0, g1x2 ϭ 1x ϩ 3

60. f 1x2 ϭ x2 ϩ 8; x Ն 0, g1x2 ϭ 1x Ϫ 8

Find the inverse function using diagrams similar to

those illustrated in Example 2. Check the result using

three test points.

31. f 1x2 ϭ x ϩ 5

32. g1x2 ϭ x Ϫ 4

4

33. p1x2 ϭ Ϫ x

5

34. r1x2 ϭ

35. f 1x2 ϭ 4x ϩ 3

3x

4

36. g1x2 ϭ 5x Ϫ 2

3

37. t1x2 ϭ 2

xϪ4

3

38. s1x2 ϭ 2

xϩ2

State the domain and range of f (x), then use the

algebraic method to find the inverse function and state

its domain and range. Finally, find any three ordered

pairs (a, b) that satisfy f, and verify the ordered pairs

(b, a) satisfy the equation for f ؊1.

3

39. f 1x2 ϭ 2

xϪ2

3

40. f 1x2 ϭ 2

xϩ3

8

43. f 1x2 ϭ

xϩ2

12

44. f 1x2 ϭ

xϪ1

x

45. f 1x2 ϭ

xϩ1

xϩ2

46. f 1x2 ϭ

1Ϫx

41. f 1x2 ϭ x3 ϩ 1

42. f 1x2 ϭ x3 Ϫ 2

The functions given in Exercises 47 through 52 are not

one-to-one. (a) Determine a domain restriction that

preserves all range values and creates a one-to-one

function, then state the new domain and range. (b) State

the domain and range of the inverse function and find

its equation.

47. f 1x2 ϭ 1x ϩ 52

49. v1x2 ϭ

2

8

1x Ϫ 32 2

51. p1x2 ϭ 1x ϩ 42 2 Ϫ 2

48. g1x2 ϭ x ϩ 3

2

50. V1x2 ϭ

52. q1x2 ϭ

4

ϩ2

x2

4

ϩ1

1x Ϫ 22 2

Find the inverse of each function f(x) given, then prove

(by composition) your inverse function is correct. Note the

domain and range of f in each case is all real numbers.

61. f 1x2 ϭ 3x Ϫ 5

62. f 1x2 ϭ 5x ϩ 4

63. f 1x2 ϭ

64. f 1x2 ϭ

xϪ5

2

xϩ4

3

65. f 1x2 ϭ 12x Ϫ 3

66. f 1x2 ϭ 23x ϩ 1

3

69. f 1x2 ϭ 2

2x ϩ 1

3

70. f 1x2 ϭ 2

3x Ϫ 2

67. f 1x2 ϭ x3 ϩ 3

71. f 1x2 ϭ

1x Ϫ 12 3

8

68. f 1x2 ϭ x3 Ϫ 4

72. f 1x2 ϭ

1x ϩ 32 3

Ϫ27

The functions given in Exercises 73 through 78 are oneto-one. State the implied domain of each function given,

and use these to state the domain and range of the

inverse function. Then find the inverse and prove by

composition that your inverse is correct.

73. f 1x2 ϭ 13x ϩ 2

74. g1x2 ϭ 12x Ϫ 5

75. p1x2 ϭ 2 1x Ϫ 3

76. q1x2 ϭ 41x ϩ 1

77. v1x2 ϭ x ϩ 3; x Ն 0

78. w1x2 ϭ x2 Ϫ 1; x Ն 0

2

Determine the domain and range for each one-to-one

function whose graph is given, and use this information

to state the domain and range of the inverse function.

Then sketch in the line y ‫ ؍‬x, estimate the location of

two or more points on the graph, and use this

information to graph f ؊1(x) on the same grid.

79.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

80.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

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81.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

82.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

83.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

1 2 3 4 5 x

Ϫ2

Ϫ3

Ϫ4

Ϫ5

84.

y

5

4

3

2

1

Ϫ5Ϫ4Ϫ3Ϫ2Ϫ1

Ϫ1

Ϫ2

Ϫ3

Ϫ4

Ϫ5

1 2 3 4 5 x

For the functions given, (a) find f ؊1(x), then use your calculator to verify they are inverses by (b) using ordered pairs,

(c) composing the functions, and (d) showing their graphs are symmetric to y ‫ ؍‬x.

85. f 1x2 ϭ 2x ϩ 1

87. h1x2 ϭ

x

xϩ1

88. j1x2 ϭ 2 2x ϩ 9 Ϫ 6

WORKING WITH FORMULAS

89. The height of a projected image: f (x) ‫ ؍‬12x ؊ 8.5

The height of an image projected on a screen

is given by the formula shown, where f 1x2

represents the actual height of the image on the

projector (in centimeters) and x is the distance of

the projector from the screen (in centimeters).

(a) When the projector is 80 cm from the screen,

how large is the image? (b) Show that the inverse

function is f Ϫ1 1x2 ϭ 2x ϩ 17, then input your

answer from part (a) and comment on the result.

What information does the inverse function

give?

86. g1x2 ϭ x2 ϩ 1; x Ն 0

90. The radius of a sphere: r(x) ‫؍‬

3x

B4␲

3

In generic form, the radius of a sphere is given by

the formula shown, where r(x) represents the radius

and x represents the volume of the sphere in cubic

units. (a) If a weather balloon that is roughly

spherical holds 14,130 in3 of helium, what is the

radius of the balloon (use ␲ Ϸ 3.142 ? (b) Show that

the inverse function is rϪ1 1x2 ϭ 43␲x3, then input

result. What information does the inverse function

give?

APPLICATIONS

91. Temperature and altitude: The temperature (in degrees Fahrenheit) at a given altitude can be approximated by

the function f 1x2 ϭ Ϫ72x ϩ 59, where f(x) represents the temperature and x represents the altitude in thousands of

feet. (a) What is the approximate temperature at an altitude of 35,000 ft (normal cruising altitude for commercial

airliners)? (b) Find f Ϫ1 1x2, and state what the independent and dependent variables represent. (c) If the

temperature outside a weather balloon is Ϫ18°F, what is the approximate altitude of the balloon?

92. Fines for speeding: In some localities, there is a set formula to determine the amount of a fine for exceeding

posted speed limits. Suppose the amount of the fine for exceeding a 50 mph speed limit was given by the

function f 1x2 ϭ 12x Ϫ 560 (x 7 50) where f (x) represents the fine in dollars for a speed of x mph. (a) What is

the fine for traveling 65 mph through this speed zone? (b) Find f Ϫ1 1x2, and state what the independent and

dependent variables represent. (c) If a fine of \$172 were assessed, how fast was the driver going through this

speed zone?

93. Effect of gravity: Due to the effect of gravity, the distance an object has fallen after being dropped is given by

the function f 1x2 ϭ 16x2; x Ն 0, where f (x) represents the distance in feet after x sec. (a) How far has the object

fallen 3 sec after it has been dropped? (b) Find f Ϫ1 1x2, and state what the independent and dependent variables

represent. (c) If the object is dropped from a height of 784 ft, how many seconds until it hits the ground (stops

falling)?

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94. Area and radius: In generic form, the area of a

circle is given by f 1x2 ϭ ␲x2, where f (x) represents

the area in square units for a circle with radius x.

(a) A pet dog is tethered to a stake in the backyard.

If the tether is 10 ft long, how much area does the

dog have to roam (use ␲ Ϸ 3.14)? (b) Find f Ϫ1 1x2,

and state what the independent and dependent

variables represent. (c) If the owners want to allow

the dog 1256 ft2 of area to live and roam, how long

a tether should be used?

95. Volume of a cone: In

r

generic form, the volume

of an equipoise cone

h

hϭr

1

3

given by f 1x2 ϭ 3␲x ,

where f(x) represents

the volume in units3 and

x represents the height of the cone. (a) Find the

volume of such a cone if r ϭ 30 ft (use ␲ Ϸ 3.142.

(b) Find f Ϫ1 1x2, and state what the independent and

491

dependent variables represent. (c) If the volume of

water in the cone is 763.02 ft3, how deep is the

water at its deepest point?

96. Wind power: The power delivered by a certain

wind-powered generator can be modeled by the

x3

, where f (x) is the horsepower

function f 1x2 ϭ

2500

(hp) delivered by the generator and x represents

the speed of the wind in miles per hour. (a) Use

the model to determine how much horsepower is

generated by a 30 mph wind. (b) The person

monitoring the output of the generators (wind

generators are usually erected in large numbers)

would like a function that gives the wind speed

based on the horsepower readings on the gauges

in the monitoring station. For this purpose, find

f Ϫ1 1x2 and state what the independent and dependent

variables represent. (c) If gauges show 25.6 hp is

being generated, how fast is the wind blowing?

EXTENDING THE CONCEPT

97. For a deeper understanding of the algebraic method

for finding an inverse, suppose a function f is

defined as f 1x2:5 1x, y2 | y ϭ 3x Ϫ 66. We can then

define the inverse as f Ϫ1: 5 1x, y2 | x ϭ 3y Ϫ 66,

having interchanged x and y in the equation

portion. The equation for f Ϫ1 is not in standard

form, but (x, y) still represents all ordered pairs

satisfying either equation. Solving for y gives

x

f Ϫ1: e 1x, y2 ` y ϭ ϩ 2 f , and demonstrates the

3

role of steps 2, 3, and 4 of the method. (a) Find five

ordered pairs that satisfy the equation for f, then

(b) interchange their coordinates and show they

satisfy the equation for f Ϫ1.

Section 5.1 One-to-One and Inverse Functions

98. By inspection, which of the following is the inverse

2

1 5 4

function for f 1x2 ϭ ax Ϫ b ϩ ?

3

2

5

a. f Ϫ1 1x2 ϭ

2

4

5 1

ax Ϫ b Ϫ

3

5

B2

3 5

5

b. f Ϫ1 1x2 ϭ 2 1x Ϫ 22 Ϫ

2

4

3 5

1

5

ax ϩ b Ϫ

c. f Ϫ1 1x2 ϭ

2B

2

4

4

1

5 3

ax Ϫ b ϩ

d. f Ϫ1 1x2 ϭ

B2

5

2

99. (R.3) Write as many of the following formulas as you can from memory:

a. perimeter of a rectangle

d. volume of a cone

b. area of a circle

e. circumference of a circle

c. volume of a cylinder

f. area of a triangle

100. (3.2) Given f 1x2 ϭ x2 Ϫ x Ϫ 2, solve the

inequality f 1x2 Յ 0 using the x-intercepts and endbehavior of the graph.

101. (3.4) For the function y ϭ 2 1x ϩ 3, find the

average rate of change between x ϭ 1 and x ϭ 2,

and between x ϭ 4 and x ϭ 5. Which is greater?

Why?

g. area of a trapezoid

h. volume of a sphere

i. Pythagorean theorem

102. (R.4) Solve the following cubic equations by

factoring:

a. x3 Ϫ 5x ϭ 0

b. x3 Ϫ 7x2 Ϫ 4x ϩ 28 ϭ 0

c. x3 Ϫ 3x2 ϭ 0

d. x3 Ϫ 3x2 Ϫ 4x ϭ 0

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CHAPTER 5 Exponential and Logarithmic Functions

5.2

Exponential Functions

LEARNING OBJECTIVES

In Section 5.2 you will see

how we can:

A. Evaluate an exponential

Demographics is the statistical study of human populations. In this section, we introduce the family of exponential functions, which are widely used to model population

growth or decline with additional applications in science, engineering, and many other

fields. As with other functions, we begin with a study of the graph and its characteristics.

function

B. Graph general

exponential functions

C. Graph base-e

exponential functions

D. Solve exponential

equations and

applications

A. Evaluating Exponential Functions

In the boomtowns of the old west, it was not

uncommon for a town to double in size every year

(at least for a time) as the lure of gold drew more

and more people westward. When this type of

growth is modeled using mathematics, exponents

play a lead role. Suppose the town of Goldsboro

had 1000 residents when gold was first discovered.

After 1 yr the population doubled to 2000 residents. The next year it doubled again to 4000, then

again to 8000, then to 16,000 and so on. You probably recognize the digits in blue as

powers of two (indicating the population is doubling), with each one multiplied by 1000

(the initial population). This suggests we can model the relationship using

P1x2 ϭ 1000 # 2x

where P(x) is the population after x yr. Further, we can evaluate this function, called

an exponential function, for fractional parts of a year using rational exponents. The

population of Goldsboro one-and-a-half years after the gold rush was

3

3

P a b ϭ 1000 # 22

2

ϭ 1000 # 1 122 3

Ϸ 2828 people

In general, exponential functions are defined as follows.

WORTHY OF NOTE

To properly understand the

exponential function and its graph

requires that we evaluate f 1x2 ϭ 2x

even when x is irrational. For

example, what does 215 mean?

While the technical details require

calculus, it can be shown that

successive approximations of 215

as in 22.2360, 22.23606, 22.236067, . . .

approach a unique real number,

and that f 1x2 ϭ 2x exists for all real

numbers x.

Exponential Functions

For b 7 0, b

1, and all real numbers x,

f 1x2 ϭ bx

defines the base b exponential function.

Limiting b to positive values ensures that outputs will be real numbers, and the

restriction b 1 is needed since y ϭ 1x is a constant function (1 raised to any power

is still 1). Specifically note the domain of an exponential function is all real numbers,

and that all of the familiar properties of exponents still hold. A summary of these properties follows. For a complete review, see Section R.2.

Exponential Properties

For real numbers a, b, m, and n, with a, b 7 0,

bm # bn ϭ bmϩn

1ab2 n ϭ an # bn

492

bm

ϭ bmϪn

bn

1

bϪn ϭ n

b

1bm 2 n ϭ bmn

b Ϫn

a n

a b ϭa b

a

b

5–14

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Section 5.2 Exponential Functions

EXAMPLE 1

493

Evaluating Exponential Functions

Evaluate each exponential function for x ϭ 2, x ϭ Ϫ1, x ϭ 12, and x ϭ ␲. Use a

calculator for x ϭ ␲, rounding to five decimal places.

4 x

a. f 1x2 ϭ 4x

b. g1x2 ϭ a b

9

Solution

a. For

f 1x2 ϭ 4x,

f 122 ϭ 42 ϭ 16

f 1Ϫ12 ϭ 4Ϫ1 ϭ

1

4

4 x

b. For g1x2 ϭ a b ,

9

4 2 16

g122 ϭ a b ϭ

9

81

4 Ϫ1 9

g1Ϫ12 ϭ a b ϭ

9

4

1

1

1

f a b ϭ 42 ϭ 14 ϭ 2

2

A. You’ve just seen how we

can evaluate an exponential

function

4

2

1

4 2

ϭ

ga b ϭ a b ϭ

A9

2

9

3

4

g1␲2 ϭ a b Ϸ 0.07827

9

f 1␲2 ϭ 4␲ Ϸ 77.88023

Now try Exercises 7 through 10

B. Graphing Exponential Functions

To gain a better understanding of exponential functions, we’ll graph examples of

y ϭ bx and note some of the characteristic features. Since b 1, it seems reasonable

that we graph one exponential function where b 7 1 and one where 0 6 b 6 1.

EXAMPLE 2

Graphing Exponential Functions with b Ͼ 1

Graph y ϭ 2x using a table of values.

Solution

To get an idea of the graph’s shape we’ll use integer values from Ϫ3 to 3 in our

table, then draw the graph as a continuous curve, since the function is defined for

all real numbers.

y

x

y ‫ ؍‬2x

Ϫ3

2Ϫ3 ϭ 18

Ϫ2

2Ϫ2 ϭ 14

Ϫ1

2Ϫ1 ϭ 12

0

20 ϭ 1

1

2 ϭ2

2

22 ϭ 4

3

23 ϭ 8

(3, 8)

8

4

(2, 4)

(1, 2)

1

(0, 1)

Ϫ4

4

x

WORTHY OF NOTE

As in Example 2, functions that are

increasing for all x ʦ D are said to

be monotonically increasing or

simply monotonic functions. The

function in Example 3 is monotonically decreasing.

Now try Exercises 11 and 12

Several important observations can now be made. First note the x-axis (the line

y ϭ 0) is a horizontal asymptote for the function, because as x S Ϫq, y S 0. Second,

the function is increasing over its entire domain, giving the function a range of

y ʦ 10, q 2.

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CHAPTER 5 Exponential and Logarithmic Functions

EXAMPLE 3

Solution

Graphing Exponential Functions with 0 Ͻ b Ͻ 1

Graph y ϭ 1 12 2 x using a table of values.

Using properties of exponents, we can write 1 12 2 x as 1 21 2 Ϫx ϭ 2Ϫx. Again using

integers from Ϫ3 to 3, we plot the ordered pairs and draw a continuous curve.

y

x

y ‫ ؍‬2؊x

Ϫ3

2Ϫ1Ϫ32 ϭ 23 ϭ 8

Ϫ2

2Ϫ1Ϫ22 ϭ 22 ϭ 4

Ϫ1

2Ϫ1Ϫ12 ϭ 21 ϭ 2

(Ϫ3, 8)

(Ϫ2, 4)

20 ϭ 1

0

Ϫ1

ϭ

1

2

2

2Ϫ2 ϭ

3

2Ϫ3 ϭ

8

4

(Ϫ1, 2)

(0, 1)

1

2

1

4

1

8

Ϫ4

x

4

Now try Exercises 13 and 14

We note this graph is also asymptotic to the x-axis, but decreasing on its domain.

In addition, both y ϭ 2x and y ϭ 2Ϫx ϭ 1 12 2 x have a y-intercept of (0, 1) and both are

one-to-one, which suggests that an inverse function can be found. Finally, observe that

y ϭ bϪx is a reflection of y ϭ bx across the y-axis, a property that indicates these basic

graphs might also be transformed in other ways, as were the toolbox functions. The

characteristics of exponential functions are summarized here:

f 1x2 ‫ ؍‬bx, b Ͼ 0 and b

1

• one-to-one function

• y-intercept (0, 1)

• domain: x ʦ ‫ޒ‬

• range: y ʦ 10, q 2

• increasing if b 7 1

• decreasing if 0 6 b 6 1

• asymptotic to the x-axis (the line y ϭ 0)

Figure 5.18

Figure 5.19

y

y

f(x) ϭ bx

0ϽbϽ1

f(x) ϭ

bϾ1

bx

WORTHY OF NOTE

When an exponential function is

increasing, it can be referred to as a

“growth function.” When decreasing,

it is often called a “decay function.”

Each of the graphs shown in

Figures 5.18 and 5.19 should now

basic functions, to be sketched

from memory and analyzed or used

as needed.

(1, b)

(0, 1)

(0, 1)

Ϫ4

4

x

Ϫ4

΂1, 1b΃

4

x

Just as the graph of a quadratic function maintains its parabolic shape regardless

of the transformations applied, exponential functions will also maintain their general shape and features. Any sum or difference applied to the basic function

1y ϭ bx Ϯ k vs. y ϭ bx 2 will cause a vertical shift in the same direction as the sign, and

any change to input values 1y ϭ bxϩh vs. y ϭ bx 2 will cause a horizontal shift in a

direction opposite the sign. For cases where multiple transformations are to be

applied, refer to the sequence outlined on page 209.

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Section 5.2 Exponential Functions

EXAMPLE 4

Graphing Exponential Functions Using Transformations

Graph F1x2 ϭ 2xϪ1 ϩ 2 using transformations of

the basic function f 1x2 ϭ 2x (not by simply plotting

points). Clearly state what transformations are

applied.

Solution

y

F(x) = 2x is shifted

1 unit right

2 units up

The graph of F is that of the basic function

f 1x2 ϭ 2x with a horizontal shift 1 unit right and

a vertical shift 2 units up. With this in mind the

horizontal asymptote also shifts from y ϭ 0 to

y ϭ 2 and (0, 1) shifts to (1, 3). The y-intercept of

F is at (0, 2.5):

(0, 2.5)

(3, 6)

(1, 3)

yϭ2

Ϫ4

4

x

F102 ϭ 2102Ϫ1 ϩ 2

ϭ 2Ϫ1 ϩ 2

1

ϭ ϩ2

2

ϭ 2.5

B. You’ve just seen how

we can graph general

exponential functions

To help sketch a more accurate graph, the point (3, 6) can be used: F132 ϭ 6.

Now try Exercises 15 through 30

C. The Base-e Exponential Function: f(x) ‫ ؍‬ex

In nature, exponential growth occurs when the rate of change in a population’s growth

is in constant proportion to its current size. Using the rate of change notation,

¢P

ϭ kP, where k is a constant. For the city of Goldsboro, we know the population at

¢t

time t is given by P1t2 ϭ 1000 # 2t, but have no information on this value of k (see

Exercise 90). We can actually rewrite this function, and other exponential functions,

using a base that gives the value of k directly and without having to apply the difference quotient. This new base is an irrational number, symbolized by the letter e. In

Section 5.6 we’ll develop the number e in the context of compound interest, while

making numerous references to our discussion here, where we define e as follows.

WORTHY OF NOTE

Just as the ratio of a circle’s

circumference to its diameter is an

irrational number symbolized by ␲,

the irrational number that results

1 x

from a1 ϩ b for infinitely large

x

x is symbolized by e. Writing

exponential functions in terms of e

simplifies many calculations in

of exponential functions.

The Number e

For x 7 0,

1 x

as x S q, a1 ϩ b S e

x

1 x

In words, e is the number that a1 ϩ b approaches as x becomes infinitely large.

x

1 x

It has been proven that as x grows without bound, a1 ϩ b indeed approaches the

x

unique, irrational number that we have named e. Table 5.5 gives approximate values of

the expression for selected values of x, and shows e Ϸ 2.71828 to five decimal places.

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CHAPTER 5 Exponential and Logarithmic Functions

The result is the base-e exponential

function: f 1x2 ϭ ex, also called the natural

exponential function. Instead of having to

enter a decimal approximation when computing with e, most calculators have an “ex ” key,

usually as the 2nd function for the key

marked LN . To find the value of e2, use the

keystrokes 2nd LN 2 )

, and the calculator display should read 7.389056099. Note

the calculator supplies the left parenthesis for

the exponent, and you must supply the right.

See Figure 5.20.

Figure 5.20

Table 5.5

10

Solution

2.59

2.705

100

1000

ENTER

EXAMPLE 5

Approximate Value

11 ؉ 1x 2 x

x

2.7169

2.71815

10,000

100,000

2.718268

2.7182805

1,000,000

10,000,000

2.71828169

Evaluating the Natural Exponential Function

Use a calculator to evaluate f 1x2 ϭ ex for the values of x given. Round to six

decimal places as needed.

a. f (3)

b. f (1)

c. f (0)

d. f 1 12 2

a. f 132 ϭ e3 Ϸ 20.085537

c. f 102 ϭ e0 ϭ 1 (exactly)

b. f 112 ϭ e1 Ϸ 2.718282

1

d. f 1 12 2 ϭ e2 Ϸ 1.648721

Now try Exercises 31 through 36

C. You’ve just seen

how we can graph base-e

exponential functions

Although e is an irrational number, the graph

of y ϭ ex behaves in exactly the same way and has

the same characteristics as other exponential graphs.

Figure 5.21 shows this graph on the same grid as

y ϭ 2x and y ϭ 3x. As we might expect, all three

graphs are increasing, have an asymptote at y ϭ 0,

and contain the point (0, 1), with the graph of y ϭ ex

“between” the other two. The domain for all three

functions, as with all basic exponential functions, is

x ʦ 1Ϫq, q 2 with range y ʦ 10, q 2. The same

transformations applied earlier can also be applied to

the graph of y ϭ ex. See Exercises 37 through 42.

Figure 5.21

10

y y ϭ 3x

9

8

y ϭ 2x

7

6

5

y ϭ ex

4

3

2

1

Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1

Ϫ1

1

2

3

4

5

x

Ϫ2

D. Solving Exponential Equations Using

the Uniqueness Property

Since exponential functions are one-to-one, we can solve equations where each side is

an exponential term with the identical base. This is because one-to-oneness guarantees

a unique solution to the equation.

WORTHY OF NOTE

Exponential functions are very

different from the power functions

studied earlier. For power functions,

the base is variable and the

exponent is constant: y ϭ xb, while

for exponential functions the

exponent is a variable and the base

is constant: y ϭ bx.

Exponential Equations and the Uniqueness Property

For all real numbers m, n, and b, where b 7 0 and b

1.

If b ϭ b ,

then m ϭ n.

m

n

2.

1,

If m

then bm

Equal bases imply exponents are equal.

n,

bn

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