Ebook College algebra (9th edition): Part 2

Ebook College algebra (9th edition): Part 2 includes the following content: Chapter 6 exponential and logarithmic functions; chapter 7 analytic geometry; chapter 8 systems of equations and inequalities; chapter 9 sequences; induction; the binomial theorem; chapter 10 counting and probability; appendix: graphing utilities. Please refer to the documentation for more details.

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—See the Internet-based Chapter Project I—

Until now, our study of functions has concentrated on polynomial and rationalfunctions These functions belong to the class of algebraic functions, that is, functions that can be expressed in terms ofsums, differences, products, quotients, powers, or roots of polynomials Functions that are not algebraic are termedtranscendental (they transcend, or go beyond, algebraic functions)

In this chapter, we study two transcendental functions: the exponential functionand the logarithmic function These functions occur frequently in a wide variety of applications, such as biology, chemistry,economics, and psychology

The chapter begins with a discussion of composite, one-to-one, and inverse functions, concepts needed to see therelationship between exponential and logarithmic functions

Exponential and Logarithmic Functions

6.9 Building Exponential, Logarithmic,and Logistic Models from Data

• Chapter Review

• Chapter Test

• Cumulative Review

• Chapter Projects

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SECTION 6.1 Composite Functions 401

Figure 1

Look carefully at Figure 2 Only those x’s in the domain of for which is inthe domain of can be in the domain of The reason is that if is not in thedomain of then is not defined Because of this, the domain of is asubset of the domain of g;the range of f ⴰ gis a subset of the range of f

f ⴰ g

f1g1x22f

g1x2

f ⴰ g

f

g1x2g

Form a Composite Function

Suppose that an oil tanker is leaking oil and you want to determine the area of thecircular oil patch around the ship See Figure 1 It is determined that the oil isleaking from the tanker in such a way that the radius of the circular patch of oil

around the ship is increasing at a rate of 3 feet per minute Therefore, the radius r of the oil patch at any time t, in minutes, is given by So after 20 minutes the

The area A of a circle as a function of the radius r is given by Thearea of the circular patch of oil after 20 minutes is square

is the output a function!

In general, we can find the area of the oil patch as a function of time t by

is a special type of function called a composite function.

obtain the original function:

In general, suppose that and are two functions and that x is a number in the

domain of By evaluating at x, we get If is in the domain of then

we may evaluate at and obtain the expression The correspondence

from x to f1g1x22fis called a composite functiong1x2 f ⴰ g.f1g1x22.

f,

g1x2

gg

gf

as “ composed with ”), is defined by

The domain of is the set of all numbers x in the domain of such that

is in the domain of f

1f ⴰ g21x2 = f1g1x22

gf

f ⴰ gg,

f

Range of g

Domain of f Domain of g

f ° g

g x

Now Work the ‘Are You Prepared?’problems on page 406

OBJECTIVES 1 Form a Composite Function (p 401)

2 Find the Domain of a Composite Function (p 402)

6.1 Composite Functions

• Find the Value of a Function (Section 3.1, pp 203–206) • Domain of a Function (Section 3.1, pp 206–208)

PREPARING FOR THIS SECTION Before getting started, review the following:

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402 CHAPTER 6 Exponential and Logarithmic Functions

Figure 3 provides a second illustration of the definition Here x is the input to

the function yielding Then is the input to the function yielding

Notice that the “inside” function in g f1g1x22is done first

f(-2) = 2(-2)2 - 3 = 5

c1f ⴰ f21-22 = f1f1-222 = f152 = 2#52 - 3 = 47

f(1) = -1

g(x) = 4x

f(x) = 2x c2 - 3 c1g ⴰ f2112 = g1f1122 = g1-12 = 4#1-12 = -4

Since the domains of both and are the set of all real numbers, the domain of

is the set of all real numbers

Solution

* Consult your owner’s manual for the appropriate keystrokes.

Evaluating a Composite Function

(a) 1f ⴰ g2112 (b) 1g ⴰ f2112 (c) 1f ⴰ f21-22 (d) 1g ⴰ g21-12

g1x2 = 4x

f1x2 = 2x2 - 3

E X A M P L E 1

Finding a Composite Function and Its Domain

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(b)

Since the domains of both and are the set of all real numbers, the domain of

is the set of all real numbers

Look back at Figure 2 on page 401 In determining the domain of the composite

the input x.

1 Any x not in the domain of must be excluded.

2 Any x for which g1x2is not in the domain of must be excluded.f

g1f ⴰ g21x2 = f1g1x22,

= 2x2 + 6x - 2 + 3 = 2x2 + 6x + 1

g(x) = 2x + 3c1g ⴰ f21x2 = g1f1x22 = g1x2 + 3x - 12 = 21x2 + 3x - 12 + 3

means that cannot equal Solve the equation to determine what

additional value(s) of x to exclude.

Also exclude from the domain of The domain of is

5x ƒ x Z 16,g

Finding the Domain of

Finding a Composite Function and Its Domain

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We could also find the domain of by first looking at the domain of

We exclude 1 from the domain of as a result Then we look

at and notice that x cannot equal since results in division by 0

So we also exclude from the domain of Therefore, the domain of

is

1 + 21x + 22 =

x + 22x + 5

x = -52

Substitute into the rule for ,

= x + 4 - 4 = x

f(x) = 3x - 4.

f g(x)

Showing That Two Composite Functions Are Equal

for every x in the domain of f ⴰ gand g ⴰ f

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Substitute into the rule for ,

Some techniques in calculus require that we be able to determine the components

because H1x2 = 1f ⴰ g21x2 = f1g1x22 = f1x + 12 = 1x + 1.f g, f1x2 = 1x g1x2 = x + 1,

H1x2 = 1 x + 1

1f ⴰ g21x2 = 1g ⴰ f21x2 = x

gf1f ⴰ g21x2 = 1g ⴰ f21x2 = x

= 1

3313x - 42 + 44

f(x) = 3x - 4

= g13x - 421g ⴰ f21x2 = g1f1x22

The function H takes and raises it to the power 50 A natural way to

decom-pose H is to raise the function to the power 50 If we let and then

See Figure 5

Other functions and may be found for which in Example 6 For

Although the functions and found as a solution to Example 6 are not unique,there is usually a “natural” selection for and that comes to mind first.f g

gf1f ⴰ g21x2 = f1g1x22 = f11x2 + 12252 = 31x2 + 122542 = 1x2 + 1250

g1x2 = 1x2 + 1225

,

f1x2 = x2

f ⴰ g = Hg

f

= 1x2 + 1250 = H1x2

= f1x2 + 121f ⴰ g21x2 = f1g1x22

Seeing the Concept

Using a graphing calculator, let

Using the viewing window

graph only and What

do you see? TRACE to verify that Y3 = Y4

Finding the Components of a Composite Function

Find functions and such that if H1x2 = 1x2 + 1250

f ⴰ g = Hg

f

E X A M P L E 6

Finding the Components of a Composite Function

Find functions and such that if H1x2 = 1

x + 1.

f ⴰ g = Hg

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Skill Building

In Problems 7 and 8, evaluate each expression using the values given in the table.

‘Are You Prepared?’ Answers are given at the end of these exercises If you get a wrong answer, read the pages listed in red

4 Given two functions f and g, the ,

6 True or False The domain of the composite function

is the same as the domain of g1x2

1f ⴰ g21x2

Concepts and Vocabulary

6.1 Assess Your Understanding

- - -

-

(e) 1g ⴰ g21-22 (f) 1f ⴰ f21-12

1g ⴰ f21021g ⴰ f21-12

1f ⴰ g21-121f ⴰ g2112

-

-In Problems 9 and 10, evaluate each expression using the graphs of and shown in the figure.

1f ⴰ g21421f ⴰ g21-12

1g ⴰ f21021g ⴰ f21-12

(7, 5) (6, 5) (5, 4) (1, 4)

(4, 2) (2, 2)

1g ⴰ f21321g ⴰ f2122

1f ⴰ g21221f ⴰ g2112

In Problems 21–28, find the domain of the composite function f ⴰ g

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23. f1x2 = x x- 1; g1x2 = -4

x + 3; g1x2 =

2x

graph of crosses the y-axis at 23.

graph of crosses the y-axis at 68.

In Problems 63 and 64, use the functions f and g to find:

(c) the domain of and of

(d) the conditions for which

meters) of a hot-air balloon is given by

where r is the radius of the balloon (in meters) If the radius

r is increasing with time t (in seconds) according to the

balloon as a function of the time t.

66 Volume of a Balloon The volume V (in cubic meters) of

the hot-air balloon described in Problem 65 is given by

If the radius r is the same function of t as in Problem 65, find the volume V as a function of the time t.

67 Automobile Production The number N of cars produced

at a certain factory in one day after t hours of operation

is given by N1t2 = 100t - 5t2, 0 … t … 10 If the cost C

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(in dollars) of producing N cars is

find the cost C as a function of the time t of operation of the

factory

68 Environmental Concerns The spread of oil leaking from a

tanker is in the shape of a circle If the radius r (in feet) of the

spread after t hours is find the area A of the

oil slick as a function of the time t.

69 Production Cost The price p, in dollars, of a certain product

and the quantity x sold obey the demand equation

Suppose that the cost C, in dollars, of producing x units is

Assuming that all items produced are sold, find the cost C as

a function of the price p.

[Hint: Solve for x in the demand equation and then form the

composite.]

70 Cost of a Commodity The price p, in dollars, of a certain

commodity and the quantity x sold obey the demand equation

Suppose that the cost C, in dollars, of producing x units is

Assuming that all items produced are sold, find the cost C as

a function of the price p.

71 Volume of a Cylinder The volume V of a right circular

cylin-der of height h and radius r is If the height is twice

the radius, express the volume V as a function of r.

72 Volume of a Cone The volume V of a right circular cone is

If the height is twice the radius, express the

C1N2 = 15,000 + 8000N, 73 Foreign Exchange Traders often buy foreign currency in

hope of making money when the currency’s value changes.For example, on June 5, 2009, one U.S dollar could purchase0.7143 Euros, and one Euro could purchase 137.402 yen.Let represent the number of Euros you can buy with

x dollars, and let represent the number of yen you can

buy with x Euros.

(a) Find a function that relates dollars to Euros

(b) Find a function that relates Euros to yen

(c) Use the results of parts (a) and (b) to find a functionthat relates dollars to yen That is, find

(d) What is

74 Temperature Conversion The function

converts a temperature in degrees Fahrenheit, F, to a ature in degrees Celsius, C The function ,converts a temperature in degrees Celsius to a temperature in

temper-kelvins, K.

(a) Find a function that converts a temperature in degreesFahrenheit to a temperature in kelvins

(b) Determine 80 degrees Fahrenheit in kelvins

75 Discounts The manufacturer of a computer is offering twodiscounts on last year’s model computer The first discount is

a $200 rebate and the second discount is 20% off the regular

price, p.

(a) Write a function f that represents the sale price if only

the rebate applies

(b) Write a function g that represents the sale price if only

the 20% discount applies

(c) Find and What does each of these functionsrepresent? Which combination of discounts represents abetter deal for the consumer? Why?

76 If and are odd functions, show that the compositefunction is also odd

77 If is an odd function and is an even function, show that

the composite functions f f ⴰ gg and g ⴰ fare both even

f ⴰ ggf

‘Are You Prepared?’ Answers

OBJECTIVES 1 Determine Whether a Function Is One-to-One (p 409)

2 Determine the Inverse of a Function Defined by a Map or a Set

of Ordered Pairs (p 411)

3 Obtain the Graph of the Inverse Function from the Graph of the Function (p 413)

4 Find the Inverse of a Function Defined by an Equation (p 414)

6.2 One-to-One Functions; Inverse Functions

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SECTION 6.2 One-to-One Functions; Inverse Functions 409

Put another way, a function is one-to-one if no y in the range is the image of more than one x in the domain A function is not one-to-one if two different

elements in the domain correspond to the same element in the range So the function

in Figure 7 is not one-to-one because two different elements in the domain, dog and cat, both correspond to 11 Figure 8 illustrates the distinction among one-to-one

functions, functions that are not one-to-one, and relations that are not functions

f

Indiana Washington South Dakota North Carolina Tennessee

State

6.2 6.1 0.8 8.3 5.8

Population (in millions)

Each x in the domain has

one and only one image

Not a one-to-one function:

y1 is the image of both

Determine Whether a Function Is One-to-One

In Section 3.1, we presented four different ways to represent a function as (1) a map,(2) a set of ordered pairs, (3) a graph, and (4) an equation For example, Figures 6and 7 illustrate two different functions represented as mappings The function inFigure 6 shows the correspondence between states and their population (in millions).The function in Figure 7 shows a correspondence between animals and lifeexpectancy (in years)

1

Dog Cat Duck Lion Pig Rabbit

Figure 7

In Words

A function is not one-to-one if

two different inputs correspond

to the same output

to two different outputs in the range That is, if and are two differentinputs of a function , then is one-to-one if f f f(x1) Z f(x2)

x2

x1

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For functions defined by an equation and for which the graph of is

known, there is a simple test, called the horizontal-line test, to determine whether

is one-to-one

ff

y = f1x2

Solution

38 42 46 55 61

Age

57 54 34 38 HDL Cholesterol

If every horizontal line intersects the graph of a function in at most onepoint, then is f one-to-one

f

The reason that this test works can be seen in Figure 9, where the horizontalline intersects the graph at two distinct points, and Since h

is the image of both and is not one-to-one Based on Figure 9,

we can state the horizontal-line test in another way: If the graph of any horizontalline intersects the graph of a function at more than one point, then is notone-to-one

ff

(b) Figure 10(b) illustrates the horizontal-line test for Because everyhorizontal line intersects the graph of exactly once, it follows that isone-to-one

gg

g1x2 = x3

f1-1, 12,

11, 12f

y = 1

f1x2 = x2

Solution

x h

Using the Horizontal-line Test

For each function, use its graph to determine whether the function is one-to-one

E X A M P L E 2

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Determining Whether a Function Is One-to-One

Determine whether the following functions are one-to-one

(a) For the following function, the domain represents the age of five males and therange represents their HDL (good) cholesterol (mg/dL)

E X A M P L E 1

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Now Work P R O B L E M 1 9

Look more closely at the one-to-one function This function is anincreasing function Because an increasing (or decreasing) function will always have

different y-values for unequal x-values, it follows that a function that is increasing

(or decreasing) over its domain is also a one-to-one function

g1x2 = x3

We will discuss how to find inverses for all four representations of functions:(1) maps, (2) sets of ordered pairs, (3) graphs, and (4) equations We begin withfinding inverses of functions represented by maps or sets of ordered pairs

Every horizontal line intersects the graph

exactly once; g is one-to-one

A horizontal line intersects the graph

twice; f is not one-to-one

3

Figure 10

State

6.2 6.1 0.8 8.3 5.8

THEOREM A function that is increasing on an interval I is a one-to-one function on I.

A function that is decreasing on an interval I is a one-to-one function on I.

Determine the Inverse of a Function Defined by a Map

or a Set of Ordered Pairs 2

there is exactly one y in the range (because is a function); and to each y in the

range of there is exactly one x in the domain (because is one-to-one).The correspondence from the range of back to the domain of is called the

inverse function of f The symbol f-1is used to denote the inverse of f.

ff

ff,

f

f,f

The function is one-to-one To find the inverse function, we interchange theelements in the domain with the elements in the range For example, the function

Solution

In Words

Suppose that we have a one-to-one

function f where the input 5

corre-sponds to the output 10 In the

inverse function , the input 10

would correspond to the output 5

f-1

Finding the Inverse of a Function Defined by a Map

Find the inverse of the following function Let the domain of the function representcertain states, and let the range represent the state’s population (in millions) Statethe domain and the range of the inverse function

E X A M P L E 3

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State 6.2

6.1 0.8 8.3 5.8

receives as input Indiana and outputs 6.2 million So the inverse receives as input6.2 million and outputs Indiana The inverse function is shown next

The domain of the inverse function is The range of the inversefunction is Washington, South Dakota, North Carolina,

If the function is a set of ordered pairs then the inverse of , denoted

is the set of ordered pairs 1y, x2

f-1,

f1x, y2,

The domain of the function is The range of the function is

The domain of the inverse function is

Look again at Figure 11 to visualize the relationship If we start with x, apply

and then apply we get x back again If we start with x, apply and then apply

we get the number x back again To put it simply, what does, undoes, and viceversa See the illustration that follows

f,

f-1,

f,Domain of f = Range of f-1 Range of f = Domain of f-1

f-1.f

f-1f

{-3,-2,-1,0,1,2,3}

1, 8, 27}

{-27, -8, -1, 0,{-27,-8,-1,0,1,8,27}

WARNING Be careful! is a symbol

for the inverse function of The used

in is not an exponent That is,

does not mean the reciprocal of ;

Figure 11

Finding the Inverse of a Function Defined

by a Set of Ordered Pairs

Find the inverse of the following one-to-one function:

State the domain and the range of the function and its inverse

51-3, -272, 1-2, -82, 1-1, -12, 10, 02, 11, 12, 12, 82, 13, 2726

E X A M P L E 4

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Consider the function which multiplies the argument x by 2 The

inverse function undoes whatever does So the inverse function of is

so undoes what did We can verify this by showing that

5x ƒ x Z 06

f-15x ƒ x Z 16

f

Solution

Obtain the Graph of the Inverse Function from the Graph

of the Function

Suppose that is a point on the graph of a one-to-one function defined by

graph of the inverse function The relationship between the point on andthe point on is shown in Figure 13 The line segment with endpoints and is perpendicular to the line and is bisected by the line (Do you see why?) It follows that the point on is the reflection about theline y = xof the point 1a, b2on f

Verifying Inverse Functions

for all x in the domain of g for all x in the domain of

for all x in the domain of f for all x in the

Verifying Inverse Functions

For what values of x is f1f-11x22 = x?

THEOREM The graph of a one-to-one function and the graph of its inverse are

symmetric with respect to the line y = x

f-1f

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Figure 14 illustrates this result Notice that, once the graph of is known, thegraph of f-1may be obtained by reflecting the graph of about the line yf = x.

Begin by adding the graph of to Figure 15(a) Since the points

and are on the graph of the points and must be on the graph of Keeping in mind that the graph of is thereflection about the line y = xof the graph of f,draw f-1.See Figure 15(b)

Find the Inverse of a Function Defined by an Equation

The fact that the graphs of a one-to-one function and its inverse function aresymmetric with respect to the line tells us more It says that we can obtain

by interchanging the roles of x and y in Look again at Figure 14 If is defined bythe equation

then is defined by the equation

The equation defines implicitly If we can solve this equation for y,

we will have the explicit form of that is,

Let’s use this procedure to find the inverse of (Since is alinear function and is increasing, we know that is one-to-one and so has an inversefunction.)

Graphing the Inverse Function

The graph in Figure 15(a) is that of a one-to-one function Draw the graph

of its inverse

y = f1x2

E X A M P L E 7

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Step 2: If possible, solve the

implicit equation for y in terms of x

to obtain the explicit form of ,

y = f -1(x)

f-1

To find the explicit form of the inverse, solve for y.

Reflexive Property; If , then Subtract 3 from both sides

Divide both sides by 2

The explicit form of the inverse is

Step 1: Replace with y In

, interchange the variables

x and y to obtain This

equation defines the inverse

function implicitly.f-1

x = f1y2

variables x and y to obtain

This equation defines the inverse f-1implicitly

x = 2y + 3

y = 2x + 3

f1x2 = 2x + 3

f1x2

Step 3: Check the result by showing

that and f-1(f (x)) = x f(f-1(x)) = x We verified that f and f are inverses in Example 5(b).

Figure 16 Note the symmetry of the graphs with respect to the line y = x

f-11x2 = 1

21x - 32

f1x2 = 2x + 3

Procedure for Finding the Inverse of a One-to-One Function

S TEP 1: In interchange the variables x and y to obtain

This equation defines the inverse function implicitly

S TEP 2: If possible, solve the implicit equation for y in terms of x to obtain the

How to Find the Inverse Function

Find the inverse of f1x2 = 2x + 3.Graph and f f-1on the same coordinate axes

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The function is not one-to-one [Refer to Example 2(a).] However, if werestrict the domain of this function to as indicated, we have a new functionthat is increasing and therefore is one-to-one As a result, the function defined by

has an inverse function,Follow the steps given previously to find

S TEP 1: In the equation interchange the variables x and y.The result is

This equation defines (implicitly) the inverse function

Finding the Inverse of a Domain-restricted Function

Find the inverse of y = f1x2 = x2if x Ú 0.Graph and ff -1

E X A M P L E 1 0

S TEP2: Solve for y.

Multiply both sides by Apply the Distributive Property

Subtract 2y from both sides; add x to both sides

x - 1 - 2

= 2x+ 1 + x - 12x+ 1 - 21x - 12 =

3x

3 = x x Z 1

Now Work P R O B L E M S 5 1 A N D 6 5

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solution for y is obtained: So

Figure 17 illustrates the graphs of f(x) = x2, x Ú 0,and f-11x2 = 1x

3 To verify that is the inverse of show that for every x in the domain of and

for every x in the domain of

4 The graphs of and f f-1are symmetric with respect to the line y = x

f-1

f1f-11x22 = xf

f-11f1x22 = xf,

5 If and are two different inputs of a function then f is

one-to-one if

6 If every horizontal line intersects the graph of a function at

no more than one point, is a(n) function

7 If f is a one-to-one function and , then

f-1(8) =

f(3) = 8f

f

f,

x2

x1

function? Why or why not? (pp 200–208)

2 Where is the function increasing? Where is it

10 True or False If and are inverse functions, the domain

of is the same as the range of g.f

gf

f-1,

34, q2,f

f-1f

f,

f-1

Concepts and Vocabulary

In Problems 11–18, determine whether the function is one-to-one.

Dave John Chuck

Karla Debra Dawn Phoebe Range

Dave John Chuck

Karla Debra Phoebe

Range

15. 512, 62, 1-3, 62, 14, 92, 11, 1026 16. 51-2, 52, 1-1, 32, 13, 72, 14, 1226

17. 510, 02, 11, 12, 12, 162, 13, 8126 18. 511, 22, 12, 82, 13, 182, 14, 3226

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In Problems 25–32, find the inverse of each one-to-one function State the domain and the range of each inverse function.

Source: Information Please Almanac

Title

Domestic Gross (in millions)

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3

1 2

Applications and Extensions

73 Use the graph of given in Problem 43 to evaluate

76 If and g is one-to-one, what is

77 The domain of a one-to-one function f is and itsrange is State the domain and the range of

78 The domain of a one-to-one function f is and itsrange is [5, q2.State the domain and the range of f[0, q2,-1

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In applications, the symbols used for the independent and dependent variables are often based on common usage So, rather than using

to represent a function, an applied problem might use to represent the cost C of manufacturing q units of a good since,

in economics, q is used for output Because of this, the inverse notation used in a pure mathematics problem is not used when finding inverses of applied problems Rather, the inverse of a function such as will be So is a function that represents the cost C as a function of the output q, while is a function that represents the output q as a function of the cost C Problems 89–92 illustrate this idea.

89 Vehicle Stopping Distance Taking into account reaction

time, the distance d (in feet) that a car requires to come to a

complete stop while traveling r miles per hour is given by the

function

(a) Express the speed r at which the car is traveling as a

func-tion of the distance d required to come to a complete

stop

(b) Verify that is the inverse of by

(c) Predict the speed that a car was traveling if the distance

required to stop was 300 feet

90 Height and Head Circumference The head circumference C

of a child is related to the height H of the child (both in

inches) through the function

(a) Express the head circumference C as a function of

height H.

(c) Predict the head circumference of a child who is 26 inches

tall

91 Ideal Body Weight One model for the ideal body weight W

for men (in kilograms) as a function of height h (in inches) is

given by the function

(a) What is the ideal weight of a 6-foot male?

(b) Express the height h as a function of weight W.

(d) What is the height of a male who is at his ideal weight of

80 kilograms?

[Note: The ideal body weight W for women (in kilograms)

as a function of height h (in inches) is given by

]

92 Temperature Conversion The function

converts a temperature from C degrees Celsius to F degrees

(c) What is the temperature in degrees Celsius if it is

70 degrees Fahrenheit?

93 Income Taxes The function

represents the 2009 federal income tax T (in dollars) due for

a “single” filer whose modified adjusted gross income is

g dollars, where

(a) What is the domain of the function T?

(b) Given that the tax due T is an increasing linear function

of modified adjusted gross income g, find the range of the function T.

(c) Find adjusted gross income g as a function of federal income tax T What are the domain and the range of this

function?

94 Income Taxes The function

represents the 2009 federal income tax T (in dollars) due for

a “married filing jointly” filer whose modified adjusted gross

income is g dollars, where (a) What is the domain of the function T ? (b) Given that the tax due T is an increasing linear function

of modified adjusted gross income g, find the range of the function T.

(c) Find adjusted gross income g as a function of federal income tax T What are the domain and the range of this

function?

95 Gravity on Earth If a rock falls from a height of 100

meters on Earth, the height H (in meters) after t seconds is

approximately

(a) In general, quadratic functions are not one-to-one

However, the function H is one-to-one Why?

(b) Find the inverse of H and verify your result.

(c) How long will it take a rock to fall 80 meters?

79 The domain of a one-to-one function g is , and its

range is State the domain and the range of

80 The domain of a one-to-one function g is [0, 15], and its

range is (0, 8) State the domain and the range of

81 A function is increasing on the interval (0, 5)

What conclusions can you draw about the graph of

82 A function is decreasing on the interval (0, 5)

What conclusions can you draw about the graph of

83 Find the inverse of the linear function

85 A function f has an inverse function If the graph of f lies in

quadrant I, in which quadrant does the graph of lie?

86 A function f has an inverse function If the graph of f lies in

quadrant II, in which quadrant does the graph of lie?

87 The function is not one-to-one Find a suitable

restriction on the domain of f so that the new function that results is one-to-one Then find the inverse of f.

88 The function is not one-to-one Find a suitable

restriction on the domain of f so that the new function that results is one-to-one Then find the inverse of f.

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SECTION 6.3 Exponential Functions 421

98. Can a one-to-one function and its inverse be equal? What

must be true about the graph of f for this to happen? Give

some examples to support your conclusion

99. Draw the graph of a one-to-one function that contains the

points 0, 0 , and 1, 5 Now draw the graph of

its inverse Compare your graph to those of other students

Discuss any similarities What differences do you see?

100. Give an example of a function whose domain is the set of

real numbers and that is neither increasing nor decreasing on

its domain, but is one-to-one

[Hint: Use a piecewise-defined function.]

21211-2, -32,

101. Is every odd function one-to-one? Explain

102. Suppose that C g represents the cost C, in dollars, of facturing g cars Explain what 800,000 represents

manu-103. Explain why the horizontal-line test can be used to identifyone-to-one functions from a graph

21

C-12

1

Explaining Concepts: Discussion and Writing

96 Period of a Pendulum The period T (in seconds) of a

sim-ple pendulum as a function of its length l (in feet) is given by

(a) Express the length l as a function of the period T.

(b) How long is a pendulum whose period is 3 seconds?

1 Yes; for each input x there is one output y.

OBJECTIVES 1 Evaluate Exponential Functions (p 421)

2 Graph Exponential Functions (p 425)

3 Define the Number e (p 428)

4 Solve Exponential Equations (p 430)

• Quadratic Functions (Section 4.3, pp 288–296)

• Linear Functions (Section 4.1, pp 272–275)

• Horizontal Asymptotes (Section 5.2,

pp 345–346)

Evaluate Exponential Functions

In Chapter R, Section R.8, we give a definition for raising a real number a to a rational

power Based on that discussion, we gave meaning to expressions of the form

where the base a is a positive real number and the exponent r is a rational number.

But what is the meaning of where the base a is a positive real number and the exponent x is an irrational number? Although a rigorous definition requires

methods discussed in calculus, the basis for the definition is easy to follow: Select a

rational number r that is formed by truncating (removing) all but a finite number of digits from the irrational number x Then it is reasonable to expect that

ax L ar

ax,

ar

1

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It can be shown that the familiar laws for rational exponents hold for realexponents

If s, t, a, and b are real numbers with and then

Introduction to Exponential Growth

Suppose a function f has the following two properties:

1 The value of f doubles with every 1-unit increase in the independent variable x.

2 The value of f at is 5, so

We seek an equation that describes this function f The key fact is that the value of f doubles for every 1-unit increase in x.

Double the value of f at 0 to get the value at 1.Double the value of f at 1 to get the value at 2

The pattern leads us to

Most calculators have an key or a caret key for working with exponents

To evaluate expressions of the form enter the base a, then press the key(or the 冷 ¿ 冷key), enter the exponent x, and press 冷 = 冷(or 冷 ENTER 冷) 冷

Using a Calculator to Evaluate Powers of 2

Using a calculator, evaluate:

E X A M P L E 1

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where a is a positive real number , , and is a real number

The domain of is the set of all real numbers The base a is the growth factor,

and because f102 = Ca0 = C, we call C the initial value.

f

C Z 0

a Z 11a 7 02

f1x2 = Cax

In the definition of an exponential function, we exclude the base becausethis function is simply the constant function We also need to

exclude bases that are negative; otherwise, we would have to exclude many values of x

and so on, are not defined in the set of real numbers.]Finally, transformations (vertical shifts, horizontal shifts, reflections, and so on) of afunction of the form also represent exponential functions

Some examples of exponential functions are

Notice for each function that the base of the exponential expression is a constantand the exponent contains a variable

In the function , notice that the ratio of consecutive outputs isconstant for 1-unit increases in the input This ratio equals the constant 2, the base ofthe exponential function In other words,

exponent is a constant In an exponential

function, the base is a constant and

the exponent is a variable 䊏

a 7 0

f 1x2 = C#a x , aZ 1,

ax n , n Ú 2,

g 1x2 =

real number, then

Identifying Linear or Exponential Functions

Determine whether the given function is linear, exponential, or neither For thosethat are linear, find a linear function that models the data For those that areexponential, find an exponential function that models the data

E X A M P L E 2

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For each function, compute the average rate of change of y with respect to x and the

ratio of consecutive outputs If the average rate of change is constant, then the function

is linear If the ratio of consecutive outputs is constant, then the function is exponential

Solution

(a) See Table 2(a) The average rate of change for every 1-unit increase in x is -3.Therefore, the function is a linear function In a linear function the average rate

of change is the slope m, so The y-intercept b is the value of the

function at , so The linear function that models the data is

.(b) See Table 2(b) For this function, the average rate of change from to 0 is -16,and the average rate of change from 0 to 1 is -8 Because the average rate ofchange is not constant, the function is not a linear function The ratio of consecutive outputs for a 1-unit increase in the inputs is a constant, Becausethe ratio of consecutive outputs is constant, the function is an exponential function with growth factor a = 1 The initial value of the exponential function

2

12

11 7

7 4

¢y

¢x =

4 - 2

0 - (-1) = 2-1

4

8 = 12

8

16 = 12

16

32 = 12

¢y

¢x =

16 - 32

0 - (-1) = -16-1

5

3 1

-1

-12

2 5

¢y

¢x =

2 - 5

0 - (-1) = -3-1

(a)

(c) (b)

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is Therefore, the exponential function that models the data is

(c) See Table 2(c) For this function, the average rate of change from to 0 is 2,and the average rate of change from 0 to 1 is 3 Because the average rate ofchange is not constant, the function is not a linear function The ratio ofconsecutive outputs from to 0 is 2, and the ratio of consecutive outputs from 0

to 1 is Because the ratio of consecutive outputs is not a constant, the function

is not an exponential function

74

6

3 1

4(–1,12) (–3,18)

Figure 18

Graph Exponential Functions

If we know how to graph an exponential function of the form , then wecould use transformations (shifting, stretching, and so on) to obtain the graph of anyexponential function

First, we graph the exponential function f1x2 = 2x

f1x2 = ax

2

The domain of is the set of all real numbers We begin by locating somepoints on the graph of as listed in Table 3

Since for all x, the range of is From this, we conclude that the

graph has no x-intercepts, and, in fact, the graph will lie above the x-axis for all x.

As Table 3 indicates, the y-intercept is 1 Table 3 also indicates that as thevalues of get closer and closer to 0 We conclude that the x-axis is

a horizontal asymptote to the graph as This gives us the end behavior for

x large and negative.

To determine the end behavior for x large and positive, look again at Table 3 As

grows very quickly, causing the graph of to rise veryrapidly It is apparent that is an increasing function and hence is one-to-one.Using all this information, we plot some of the points from Table 3 and connectthem with a smooth, continuous curve, as shown in Figure 18

As we shall see, graphs that look like the one in Figure 18 occur very frequently

in a variety of situations For example, the graph in Figure 19 illustrates the number

Graphing an Exponential Function

Graph the exponential function: f1x2 = 2x

E X A M P L E 3

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1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

Year

Number of Cellular Phone Subscribers at Year End

220

280 260 240

Figure 19

We shall have more to say about situations that lead to exponential growth later

in this chapter For now, we continue to seek properties of exponential functions.The graph of in Figure 18 is typical of all exponential functions of theform with Such functions are increasing functions and hence are

one-to-one Their graphs lie above the x-axis, pass through the point and

asymptote There are no vertical asymptotes Finally, the graphs are smooth andcontinuous with no corners or gaps

Figure 20 illustrates the graphs of two more exponential functions whose basesare larger than 1 Notice that the larger the base, the steeper the graph is when, and when , the larger the base, the closer the graph of the equation is to

Seeing the Concept

Graph and compare what you see to Figure 18 Clear the screen and graph and and compare what you see to Figure 20 Clear the screen and graph Y1 = 10x and Y2 = 100x.

Y2 = 6x

Y1 = 3x

Y1 = 2x

Properties of the Exponential Function

1 The domain is the set of all real numbers or using intervalnotation; the range is the set of positive real numbers or usinginterval notation

2 There are no x-intercepts; the y-intercept is 1.

3 The x-axis is a horizontal asymptote as

5 The graph of contains the points and

6 The graph of is smooth and continuous, with no corners or gaps SeeFigure 21

f1x2 = ax, where a 7 1,

x: - qax = 0D

x: - q1y = 02

10,q21- q,q2

of cellular telephone subscribers at the end of each year from 1985 to 2008

We might conclude from this graph that the number of cellular telephone

subscribers is growing exponentially.

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The domain of consists of all real numbers As before, we locate some

points on the graph by creating Table 4 Since for all x, the range of is

the interval The graph lies above the x-axis and so has no x-intercepts The

values of approach 0 The x-axis is a horizontal asymptote as

It is apparent that is a decreasing function and so is one-to-one Figure 22illustrates the graph

f

x: q.1y = 02

6

3 1

4(–1,12) (–3,18)

6

3 1 2

( )1 2

We could have obtained the graph of from the graph of using

transformations The graph of is a reflection about the y-axis of

the graph of y = 2x(replace x by -x) See Figures 23(a) and (b)

Seeing the Concept

Using a graphing utility, simultaneously

Conclude that the graph of

for is the reflection about the

y-axis of the graph of Y1 = a x

Graphing an Exponential Function

Graph the exponential function: f1x2 = a1

2bx

E X A M P L E 4

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Figure 24 illustrates the graphs of two more exponential functions whose basesare between 0 and 1 Notice that the smaller base results in a graph that is steeperwhen When , the graph of the equation with the smaller base is closer

to the x-axis.

x 7 0

x 6 0

Properties of the Exponential Function

1 The domain is the set of all real numbers or using intervalnotation; the range is the set of positive real numbers or usinginterval notation

2 There are no x-intercepts; the y-intercept is 1.

3 The x-axis is a horizontal asymptote as

5 The graph of contains the points and

6 The graph of is smooth and continuous, with no corners or gaps SeeFigure 25

f1x2 = ax, 0 6 a 6 1,

C lim

x: qax = 0D

x: q1y = 02

10,q21- q,q2

( )

1, 1 2

Define the Number e

As we shall see shortly, many problems that occur in nature require the use of anexponential function whose base is a certain irrational number, symbolized by the

letter e.

3

Graphing Exponential Functions Using Transformations

Graph and determine the domain, range, and horizontal asymptote

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One way of arriving at this important number e is given next.

Table 5 illustrates what happens to the defining expression (2) as n takes on

increasingly large values The last number in the right column in the table is correct

to nine decimal places and is the same as the entry given for e on your calculator

(if expressed correctly to nine decimal places)

The exponential function whose base is the number e, occurs with such frequency in applications that it is usually referred to as the exponential

function Indeed, most calculators have the key or , which may be

used to evaluate the exponential function for a given value of x.*

冷 exp1x2 冷

f1x2 = ex,

Seeing the Concept

Graph and compare what you see to Figure 27 Use eVALUEate or TABLE to verify the points on the graph shown in Figure 27 Now graph and on the same screen as Notice that the graph of Y1 = e xlies between these two graphs.

2 e2 L 7.39

e1 L 2.72

e0 L 1

e-1 L 0.37 -

e-2 L 0.14 -

Graphing Exponential Functions Using Transformations

Graph and determine the domain, range, and horizontal asymptote

of f

f1x2 = -ex-3

E X A M P L E 6

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As Figure 28(c) illustrates, the domain of is the interval and the range is the interval The horizontal asymptote is the line

(2, ⫺e2 )

x y

⫺3

⫺6

3

(3, ⫺1) (4, ⫺e)

(5, ⫺e2 )

x y

Solve Exponential Equations

as exponential equations Such equations can sometimes be solved by appropriately

applying the Laws of Exponents and property (3):

one-to-(a) Since write the equation as

Now we have the same base, 3, on each side Set the exponents equal to eachother to obtain

The solution set is {3}

(b)

x = 11 4x - 2 = 3x + 9 2(2x - 1) = 3(x + 3)

When two exponential expressions

with the same base are equal,

then their exponents are equal

䊉

Solving Exponential Equations

Solve each exponential equation

(a) 3x+1 = 81 (b) 42x-1 = 8x+3

E X A M P L E 7

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Use the Laws of Exponents first to get a single expression with the base e on the

Use the Zero-Product Property

The solution set is 5-3, 16

x = -3 or x = 11x + 321x - 12 = 0

There is a 99.75% probability that a car will arrive within 30 minutes

(c) See Figure 29 for the graph of F.

(d) As time passes, the probability that a car will arrive increases The value that F

approaches can be found by letting Since it follows that

as We conclude that F approaches 1 as t gets large The

algebraic analysis is confirmed by Figure 29

Between 9:00 PMand 10:00 PMcars arrive at Burger King’s drive-thru at the rate of

12 cars per hour (0.2 car per minute) The following formula from statistics can be

used to determine the probability that a car will arrive within t minutes of 9:00 PM

(a) Determine the probability that a car will arrive within 5 minutes of 9 PM(that is,before 9:05 PM)

(b) Determine the probability that a car will arrive within 30 minutes of 9 PM

(before 9:30 PM)

(c) Graph F using your graphing utility.

(d) What value does F approach as t increases without bound in the positive direction?

F1t2 = 1 - e-0.2t

E X A M P L E 9

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8 True or False The domain of the exponential function

, is the set of all realnumbers

9 True or False The range of the exponential function

, is the set of all realnumbers

3 True or False To graph shift the graph of

to the left 2 units.(pp 244–253)

11 The graph of every exponential function

and , passes through three points: _, _,and _

12 If the graph of the exponential function

where , is decreasing, then a must be less

Domain: the interval range: the interval

x-intercepts: none; y-intercept: 1 Horizontal asymptote: x-axis as

Increasing; one-to-one; smooth; continuousSee Figure 21 for a typical graph

Domain: the interval range: the interval

x-intercepts: none; y-intercept: 1 Horizontal asymptote: x-axis as

Decreasing; one-to-one; smooth; continuousSee Figure 25 for a typical graph

If then au = av, u = v

x: q1y = 02

10, q21- q, q2;

f1x2 = ax, 0 6 a 6 1

x: - q1y = 02

10, q21- q, q2;

f1x2 = ax, a 7 1

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In Problems 25–32, determine whether the given function is linear, exponential, or neither For those that are linear functions, find a linear function that models the data; for those that are exponential, find an exponential function that models the data.

In Problems 41–52, use transformations to graph each function Determine the domain, range, and horizontal asymptote of each function.

3 2

2 3 -1

1 8

1 4

1 2 -1

In Problems 33–40, the graph of an exponential function is given Match each graph to one of the following functions.

2 3

⫺1

⫺2

x y

2 1

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In Problems 61–80, solve each equation.

81 If 4x = 7, what does 4-2x equal? 82 If 2x = 3, what does 4-x equal?

83 If 3-x = 2, what does 32 equal? 84 If 5-x= 3, what does 53 equal?

(2, 9)

(–1, )

20

4 8 12 16

–2

1 3

20

4 8 12 16

–2

1 5

x y

–4

–12 –8

89 Find an exponential function with horizontal asymptote

whose graph contains the points 0, 3 and 1, 5 1 2 1 2 y = 2 90 Find an exponential function with horizontal asymptotey = -3whose graph contains the points10, -22and1-2, 1 2

Mixed Practice

91 Suppose that

(a) What is ? What point is on the graph of f ?

(b) If what is x? What point is on the graph

f1x2 = 3x

93 Suppose that

(a) What is What point is on the graph of g?

of g?

g1x2 = 66,g1-12?

(a) What is What point is on the graph of g?

of g?

g1x2 = 122,g1-12?

g1x2 = 5x - 3

95 Suppose that

(a) What is What point is on the graph of H?

(a) What is What point is on the graph of F?

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In Problems 97–100, graph each function Based on the graph, state the domain and the range and find any intercepts.

97. f1x2 = eeex-x if xif x 6 0Ú 0 98. f1x2 = eeex-x if xif x 6 0Ú 0

99. f1x2 = e --eex-x if x 6 0

if x Ú 0 100. f1x2 = e --ee-xx if xif x 6 0Ú 0

101 Optics If a single pane of glass obliterates 3% of the light

passing through it, the percent p of light that passes through

n successive panes is given approximately by the function

(a) What percent of light will pass through 10 panes?

(b) What percent of light will pass through 25 panes?

102 Atmospheric Pressure The atmospheric pressure p on a

balloon or plane decreases with increasing height This

pressure, measured in millimeters of mercury, is related to

the height h (in kilometers) above sea level by the function

(a) Find the atmospheric pressure at a height of 2

kilo-meters (over a mile)

(b) What is it at a height of 10 kilometers (over 30,000 feet)?

103 Depreciation The price p, in dollars, of a Honda Civic DX

Sedan that is x years old is modeled by

(a) How much should a 3-year-old Civic DX Sedan cost?

(b) How much should a 9-year-old Civic DX Sedan cost?

104 Healing of Wounds The normal healing of wounds can be

modeled by an exponential function If represents the

original area of the wound and if A equals the area of the

wound, then the function

describes the area of a wound after n days following an

injury when no infection is present to retard the healing

Suppose that a wound initially had an area of 100 square

millimeters

(a) If healing is taking place, how large will the area of the

wound be after 3 days?

(b) How large will it be after 10 days?

105 Drug Medication The function

can be used to find the number of milligrams D of a certain

drug that is in a patient’s bloodstream h hours after the drug

has been administered How many milligrams will be

present after 1 hour? After 6 hours?

106 Spreading of Rumors A model for the number N of people

in a college community who have heard a certain rumor is

where P is the total population of the community and d is

the number of days that have elapsed since the rumor

began In a community of 1000 students, how many students

will have heard the rumor after 3 days?

107 Exponential Probability Between 12:00 PMand 1:00 PM,

cars arrive at Citibank’s drive-thru at the rate of 6 cars per

prob-will arrive within t minutes of 12:00 PM:

(a) Determine the probability that a car will arrive within

10 minutes of 12:00 PM(that is, before 12:10 PM).(b) Determine the probability that a car will arrive within

40 minutes of 12:00 PM(before 12:40 PM)

(c) What value does F approach as t becomes unbounded in

the positive direction?

(d)Graph F using a graphing utility.

(e) Using INTERSECT, determine how many minutes areneeded for the probability to reach 50%

108 Exponential Probability Between 5:00 PM and 6:00 PM,cars arrive at Jiffy Lube at the rate of 9 cars per hour(0.15 car per minute) The following formula from probabil-ity can be used to determine the probability that a car will

arrive within t minutes of 5:00 PM:

(a) Determine the probability that a car will arrive within

15 minutes of 5:00 PM(that is, before 5:15 PM)

(b) Determine the probability that a car will arrive within

30 minutes of 5:00 PM(before 5:30 PM)

(c) What value does F approach as t becomes unbounded in

the positive direction?

(d)Graph F using a graphing utility.

(e) Using INTERSECT, determine how many minutes areneeded for the probability to reach 60%

109 Poisson Probability Between 5:00 PM and 6:00 PM, carsarrive at McDonald’s drive-thru at the rate of 20 cars perhour The following formula from probability can be used to

determine the probability that x cars will arrive between

110 Poisson Probability People enter a line for the Demon

Roller Coaster at the rate of 4 per minute The following

formula from probability can be used to determine the

probability that x people will arrive within the next minute.

P1x2 = 4xe-4

x!

x = 20

x = 15x! = x#1x - 12#1x - 22#Á#3#2#1

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(a) If and

how much current is flowing after 0.3 second? After

0.5 second? After 1 second?

(b) What is the maximum current?

(c) Graph this function measuring I along the

y-axis and t along the x-axis.

(d) If and

how much current is flowing after 0.3 second? After

0.5 second? After 1 second?

(e) What is the maximum current?

(f) Graph the function on the same coordinate

(a) Determine the probability that people will arrive

within the next minute

(b) Determine the probability that people will arrive

within the next minute

111 Relative Humidity The relative humidity is the ratio

(expressed as a percent) of the amount of water vapor in the

air to the maximum amount that it can hold at a specific

temperature The relative humidity, R, is found using the

following formula:

where T is the air temperature (in °F) and D is the dew point

temperature (in °F)

(a) Determine the relative humidity if the air temperature is

50° Fahrenheit and the dew point temperature is 41°

Fahrenheit

(b) Determine the relative humidity if the air temperature is

68° Fahrenheit and the dew point temperature is 59°

Fahrenheit

(c) What is the relative humidity if the air temperature and

the dew point temperature are the same?

112 Learning Curve Suppose that a student has 500 vocabulary

words to learn If the student learns 15 words after 5 minutes,

the function

approximates the number of words L that the student will

learn after t minutes.

(a) How many words will the student learn after 30 minutes?

(b) How many words will the student learn after 60 minutes?

113 Current in a RL Circuit The equation governing the

amount of current I (in amperes) after time t (in seconds) in

a single RL circuit consisting of a resistance R (in ohms), an

inductance L (in henrys), and an electromotive force E (in

114 Current in a RC Circuit The equation governing the

amount of current I (in amperes) after time t (in microseconds)

in a single RC circuit consisting of a resistance R (in ohms), a capacitance C (in microfarads), and an electromotive force E

micro-farad, how much current is flowing initially After 1000 microseconds? After 3000 microseconds?

(b) What is the maximum current?

(c) Graph the function measuring I along the

y-axis and t along the x-axis.

micro-farads, how much current is flowing initially? After

1000 microseconds? After 3000 microseconds?

(e) What is the maximum current?

(f) Graph the function on the same coordinate axes

117 Another Formula for e Use a calculator to compute the

various values of the expression Compare the values to e.

132122112

n! = n1n - 12#Á#1! = 1, 2! = 2#1, 3! = 3#2#1,

n = 4, 6, 8,

2+ 12! + 13! + Á + 1

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SECTION 6.4 Logarithmic Functions 437

Recall that a one-to-one function has an inverse function that is defined(implicitly) by the equation In particular, the exponential function

where and is one-to-one and hence has an inversefunction that is defined implicitly by the equation

This inverse function is so important that it is given a name, the logarithmic function.

(a) Show that is an odd function

(b)Graph using a graphing utility

123 The hyperbolic cosine function, designated by cosh x, is

(b) Graph using a graphing utility

(c) Refer to Problem 122 Show that, for every x,

124 Historical Problem Pierre de Fermat (1601–1665) tured that the function

that this formula fails for Use a calculator to mine the prime numbers produced by for

deter-Then show that f152 = 641 * 6,700,417,fwhich is not prime.x = 1, 2, 3, 4.

Explaining Concepts: Discussion and Writing

125. The bacteria in a 4-liter container double every minute After

60 minutes the container is full How long did it take to fill

half the container?

126. Explain in your own words what the number e is Provide at

least two applications that use this number

127. Do you think that there is a power function that increases

more rapidly than an exponential function whose base is

greater than 1? Explain

128. As the base a of an exponential function

increases, what happens to the behavior of its graphfor What happens to the behavior of its graph for

129. The graphs of and y = a1 are identical Why?

OBJECTIVES 1 Change Exponential Statements to Logarithmic Statements and

Logarithmic Statements to Exponential Statements (p 438)

2 Evaluate Logarithmic Expressions (p 438)

3 Determine the Domain of a Logarithmic Function (p 439)

4 Graph Logarithmic Functions (p 440)

5 Solve Logarithmic Equations (p 444)

6.4 Logarithmic Functions

• Solving Inequalities (Section 1.5, pp 119–126)

• Quadratic Inequalities (Section 4.5,

pp 309–311)

• Polynomial and Rational Inequalities (Section 5.4,

pp 368–371)

• Solve Linear Equations (Section 1.1, pp 82–87)

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As this definition illustrates, a logarithm is a name for a certain exponent So,

represents the exponent to which a must be raised to obtain x.

loga x

(read as “y is the logarithm to the base a of x”) and is defined by

The domain of the logarithmic function y = loga xis x 7 0

y = loga x if and only if x = ay

Evaluate Logarithmic Expressions

To find the exact value of a logarithm, we write the logarithm in exponentialnotation using the fact that is equivalent to and use the fact that if

When you read , think to

yourself “a raised to what power

gives me x.”

loga x

Relating Logarithms to Exponents

is equivalent to the exponential statement

Changing Exponential Statements to Logarithmic Statements

Change each exponential statement to an equivalent statement involving alogarithm

E X A M P L E 2

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Changing Logarithmic Statements to Exponential Statements

Change each logarithmic statement to an equivalent statement involving anexponent

E X A M P L E 3

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Trang 40

SECTION 6.4 Logarithmic Functions 439

(a) To evaluate , think “2 raised

to what power yields 16.” So,

Change to exponentialform

Equate exponents

Therefore,

(b) To evaluate , think “3 raised

to what power yields ” So,

Change to exponentialform

y = log3

127

log3 127

Determine the Domain of a Logarithmic Function

The logarithmic function has been defined as the inverse of the

the discussion given in Section 6.2 on inverse functions, for a function and itsinverse we have

Consequently, it follows that

Domain of f-1 = Range of f and Range of f-1 = Domain of f

Domain: 0 6 x 6 q Range: - q 6 y 6 q

y = loga x 1defining equation: x = ay2

The domain of a logarithmic function consists of the positive real numbers, so

the argument of a logarithmic function must be greater than zero

(a) The domain of F consists of all x for which that is, Using

interval notation, the domain of f is

(b) The domain of is restricted to

Solving this inequality, we find that the domain of consists of all x between

and 1, that is,-1 6 x 6 1or, using interval notation,1-1, 12 -1

Finding the Exact Value of a Logarithmic Expression

Find the exact value of:

27log2 16

E X A M P L E 4

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Finding the Domain of a Logarithmic Function

Find the domain of each logarithmic function

1 - xb

F1x2 = log21x + 32

E X A M P L E 5

Tiêu đề	Exponential and Logarithmic Functions Outline
Trường học	Unknown University
Chuyên ngành	College Algebra
Thể loại	Textbook
Năm xuất bản	Unknown Year
Thành phố	Unknown City

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Số trang	401
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