6 exponential functions

Chapter 6 l Exponential Functions 233

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6.1 Inverses RevisitedFunctions, Relations, and

Their Inverses | p. 235

6.2 Let’s Try to be RationalRational Exponents | p. 243

6.3 Time to OperateSimplifying and Operating with

Radical Expressions Using Rational

Exponents | p. 249

6.4 Generic ExponentialsGraphs of Exponential

Functions | p. 257

6.5 TransformersTransformations of Exponential

Functions | p. 267

6.6 Interest and DecayExponential Functions and

Problem Solving | p. 275

Exponential Functions6CHAPTER

Georgia has two nuclear power plants: the Hatch plant in Appling County, and the Vogtle plant in

Burke County. Together, these plants supply about 22% of the state’s electricity. One byproduct

of nuclear power generation is the radioactive isotope plutonium-240, which has a half-life

(the time it takes to decay by half) of 6,560 years. You will calculate the decay characteristics

of various radioactive materials.

234 Chapter 6 l Exponential Functions

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Lesson 6.1 l Functions, Relations, and Their Inverses 235

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6.1 Inverses RevisitedFunctions, Relations, and Their Inverses

ObjectivesIn this lesson you will:

l Determine inverses of relations and

functions.

l Determine when the inverse of a function

is also a function.

l Graph functions and their inverses.

l Determine whether a function is one to

one using the horizontal line test.

Key Termsl one to one

l horizontal line test

Problem 1 Volume of a SphereThe formula for the volume of a sphere is V � 4�r 3 _____

3 , where r is the radius of

the sphere.

1. Write the volume formula as a function, f, of the radius.

2. Graph the volume function.

x43

8

24

32

–1–8

21–2

–16

–3

–24

–4

–32

y

16

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3. What is the domain of the function f(x) � 4�x 3 _____ 3 ?

4. What is the domain of the volume function in terms of the problem situation?

5. Which portion of the graph models volume? Explain.

6. Graph the volume function and the line y � x in the first quadrant. Then

sketch the inverse of the volume function.

9

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9x

y

7. Solve for the inverse of the volume function, f�1, algebraically.

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Remember that by definition a relation is a function if for every value of x there

is one and only one value of y. One method to determine whether a relation is a

function is the vertical line test. To use the vertical line test, imagine drawing every

possible vertical line on the coordinate plane. If no vertical line exists that intersects

the graph of a relation at more than one point, then the relation is a function.

8. Explain why the vertical line test is equivalent to the definition of a function.

The inverse of a function is a function if and only if the function is one to one.

A function is one to one if for every value of y there is at most one value of x.

The horizontal line test is a test to determine if a function is one to one. To use the

horizontal line test, imagine drawing every possible horizontal line on the coordinate

plane. If no horizontal line intersects the graph of a function at more than one point,

then the function is one to one.

9. Explain why the horizontal line test is equivalent to the definition of a one to

one function.

10. Is the inverse of the volume function of a sphere also a function? Explain.

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The formula for the volume of a cylinder is V � �r 2h, where r is the radius of the

base of the cylinder and h is the height.

1. Write the volume formula as a function, f, of the radius. Let the height equal

6 centimeters.

2. Graph the volume function.

x43

25

35

40

–1

15

21–2

10

–3

5

–4

20

y

30

3. What is the domain of the function f(x)?

4. What is the domain of the volume function in terms of the problem situation?

5. Which portion of the graph models volume? Explain.

Problem 2 Volume of a Cylinder

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6. Graph the volume function and the line y � x in the first quadrant.

Then sketch the inverse of the volume function.

4

3

2

1

1 2 3 4x

y

7. Solve for the inverse of the volume function, f �1, algebraically.

8. Is the function f(x) one to one? Explain.

9. Is the function in terms of the problem situation one to one? Explain.

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Determine the inverse of each function. Then, state whether the inverse is also

a function.

1. Linear

a. y � �5x � 2

b. 3y � 2x � 2 � 0

2. Quadratic

a. f(x) � x2 � 6x � 2

Problem 3 Determining Inverses Algebraically

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b. y � 2x2 � 10x � 2

3. Power

a. f(x) � 5x3 � 2

b. f(x) � 4x 4 � 2

Be prepared to share your methods and solutions.

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Lesson 6.2 l Rational Exponents 243

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6.2 Let’s Try to be RationalRational Exponents


l Evaluate expressions using rational

exponents.

l Convert between radical form and

exponential form.

Key Terml rational exponent

Problem 1 Exponents between 0 and 1

1. Graph the function f(x) � 2x for x-values between 0 and 1 using a graphing

calculator. Sketch the graph on the grid.

1

y2

x

0.20 0.4 0.6 0.8

Many x-values between 0 and 1 are rational numbers. For the function f(x) � 2x, these

rational numbers are written as exponents. But how can you evaluate an expression

with a rational number exponent? The properties of exponents can be useful.

2. Perform the following steps.

a. Solve the equation 2a � 2a � 2 for a.

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b. Let x � 2a. The equation from Question 2, part (a) can be written as x2 � 2.

Solve the equation x2 � 2.

c. Complete the following statement.

2a � 2 � x �

d. Plot the point (a, 2a) on your graph.


a. Solve for a: 2a � 2a � 2a � 2




2a � 2 � x �


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a. Solve for a: 2 1 __ 3

� 2a � 2

b. Let x � 2a. The equation from Question 4, part (a) can be written as

3

� __

2 � x � 2. Solve the equation 3

� __

2 � x � 2.


2a � 2

� x �


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a. Solve for a: 2a � 2a � 2a � 2a � 2




2a � 2 � x �


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6. Solve for a: 2 1 __ 4

� 2a � 2. Write 2a in radical form and plot the point (a, 2a) on

your graph.

7. Complete the following table.

xf(x) � 3x

Exponential Form Radical Form Numerical Value

0 30 1

1 __ 5

1 __ 3

2 __ 5

1 __ 2

3 __ 5

3 __ 4

4 __ 5

1 31 3

8. Graph the function f(x) � 3x between 0 and 1 using the table.

1

y2

x

0.20 0.4 0.6 0.8

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9. Write another property of exponents, the definition of a rational exponent, using your results.

Definition of a Rational Exponent

x a __ b

�

10. Simplify each expression using the definition of rational exponents.

a. 25 3 __ 2

b. 27 2 __ 3

c. 100 5 __ 2

d. 32 6 __ 5

e. 64 3 __ 2

f. 16 3 __ 4

11. Write each expression in radical form.

a. 2 2 __ 3

b. 5 1 __ 5

c. x 7 __ 9

d. x 2 __ 3

y 3 __ 4

12. Write each expression in exponential form.

a. √__

26 b. √__

55

c. 4

� __

x2 d. 3 � ____

x5 y3


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Lesson 6.3 l Simplifying and Operating with Radical Expressions Using Rational Exponents 249

6

6.3 Time to OperateSimplifying and Operating with Radical Expressions Using Rational Exponents


l Simplify radical and exponential expressions.

l Multiply and divide radical expressions using rational exponents.

l Rationalize the denominators of radical expressions using rational exponents.

Problem 1 Simple Radicals

One method you learned for simplifying radicals is to factor the radicand into the

appropriate roots, take the roots where possible, and multiply the remaining factors.

For example, simplify 3 � ____

x5 y7 .

3 � ____

x5 y7 � 3

� __

x3 3

� __

x2 3 � __

y3 3 � __

y3 3 � __

y1

� x � y � y 3

� __

x2 3

� __

y1

� xy2

3

� ___

x2 y

1. Simplify each radical.

a. √____

x5 y7

b. 4 � ____

x6 y9

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Rational exponents can also be used to simplify radicals. To simplify radicals using

rational exponents, convert the radical to exponential form, simplify the exponents,

and convert back to radical form. For example, simplify 3 � ____

x5 y7 .

3 � ____

x5 y7 � x 5 __ 3

y 7 __ 3

� x 3 __ 3

� x 2 __ 3

� y 6 __ 3

� y 1 __ 3

� x � x 2 __ 3

� y2 � y 1 __ 3

� xy2 3 � ___

x2 y

2. Simplify each radical using rational exponents.

a. 3 � ____

x2 y7 b. 4 � _____

x7 y11

Rational exponents can be used to multiply radical expressions. To multiply two

radical expressions using rational exponents, convert each radical to exponential

form. Then use properties of exponents to simplify the expression. Finally, write the

product in radical form and simplify. For example, multiply 3

� __

x5 � 2

� __

x3 .

3

� __

x5 � 2

� __

x3 � x 5 __ 3

x 3 __ 2

� x 5 __ 3 � 3 __

2

� x 19 ___ 6

� 6

� ___

x19

� x3 6

� ___

x19

3. Perform each multiplication by converting to rational exponents.

a. 3

� __

x5 � 4

� __

x3 b. 4

� __

x3 � √__

x5

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c. 5 � __

y4 � 3 � __

y2 d. √__

x5 � 3 � __

x � 4

� __

x3

Rational exponents can be used to divide radical expressions. To divide two radical

expressions using rational exponents, convert each radical to exponential form.

Then use properties of exponents to simplify the expression. Finally, write the

quotient in radical form and simplify.

4. Perform each division by converting to rational exponents.

a. √

__

y3 ____

3 � __

y b.

5 � __

y4 _____

3 � __

y2

c. 3

� __

x4 ____ 6

� __

x5

Rational exponents can be used to rationalize denominators of radical expressions.

For example, rationalize the denominator of the expression 2 ____ 6

� __

25 .

2 ____ 6

� __

25 � 2 __

2 5 __ 6

� 2 1 __ 6

__

2 1 __ 6

� 2 � 2 1 __ 6

______ 2

� 2 1 __ 6

� 6

� __

2

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5. Rationalize each denominator using rational exponents.

a. 3 ____ 3

� __

35 b. x ___

4 � __

x

c. x 3 � __

x _____ 4

� __

x5

6. Calculate each power using rational exponents.

a. ( √___

43x ) 2 b. ( 4 �

_______

( 16x4y3 ) ) 3

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1. Calculate each product using the properties of exponents.

a. ( 3

� ___

5x ) ( 3 � _____

12x2y )

b. ( 3

� __

x4 ) ( 3 � ____

x2y4 )

c. ( √___

x3y ) �3 ( 4 �

____

x2y3 ) 2

d. ( √___

4x ___ y2

) 3

( 12y 3

� __

x2 ) �2

Problem 2 Multiplying and Dividing Radicals

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e. ( 3 3 � ___

3y2 ______

x3 )

2

( 4

� __

x3 ____ y3

) �1

( y2

____ √

___

3x )

3

2. Calculate each quotient using the properties of exponents.

a. √

___

8x _____

3

� ___

3x

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b. 5 � ____

x2y4 ______

3 � ____

3xy

c. √

____

12x ______ 3 � ____

3x2y


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Lesson 6.4 l Graphs of Exponential Functions 257

6

Problem 1 The Graph of f(x) � bx

1. Sketch the graph of f(x) � 2x.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

2. Identify each characteristic of the function f(x) � 2x and its graph.

a. Domain

b. Range

c. Intercepts of this function

6.4 Generic ExponentialsGraphs of Exponential Functions


l Graph exponential functions.

l Identify characteristics of graphs of exponential functions.

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d. Intervals of increase or decrease

e. Asymptotes

3. Sketch the graph of g(x) � 3x.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

4. Identify each characteristic of the function g(x) � 3x and its graph.

a. Domain

b. Range



e. Asymptotes

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5. What do you notice about the characteristics for f(x) � 2x and g(x) � 3x?

6. What are the characteristics of h(x) � 5x? k(x) � 10x?

7. The graph of f(x) � bx is shown. Identify each characteristic of the

function f(x) � bx and its graph.

x43–1 21–2–3–4

y

a. Domain

b. Range



e. Asymptotes

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Problem 2 The Graph of f(x) � b�x

1. Sketch the graph of f(x) � 2�x.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

2. Identify each characteristic of the function f(x) � 2�x and its graph.

a. Domain

b. Range



e. Asymptotes

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2. Describe the similarities and differences between the graphs of f(x) � 2x and

f(x) � 2�x.

3. Sketch the graph of g(x) � 3�x.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

4. Identify each characteristic of the function g(x) � 3�x and its graph.

a. Domain

b. Range



e. Asymptotes

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5. Describe the similarities and differences between the graphs of g(x) � 3x and

g(x) � 3�x.

6. The graph of f(x) � b�x is shown. Identify each characteristic of the function

f(x) � b�x and its graph.

x43–1 21–2–3–4

y

a. Domain

b. Range



e. Asymptotes

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7. Describe the similarities and differences between the graphs of f(x) � bx and

f(x) � b�x.

Problem 3 The Graph of f(x) � abx

1. Define three functions of the form f(x) � abx with a greater than 0.

2. Sketch the graph of each function from Question 1.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

x43

2

4

6

8

–1–2

210–2

–4

–3

–6

–4

–8

y

x86

2

4

6

8

–2–2

420–4

–4

–6–8

y

10

12

14

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3. Identify the characteristics of each function and its graph.

a. Domain

b. Range



e. Asymptotes

4. The graph of f(x) � abx is shown. Identify each characteristic of the function

f(x) � abx and its graph.

x43–1 21–2–3–4

y

a. Domain

b. Range

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e. Asymptotes

f. What is the sign of the constant a?

5. The graph of f(x) � ab�x is shown. Identify each characteristic of the function

f(x) � ab�x and its graph.

x43–1 21–2–3–4

y

a. Domain

b. Range


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Lesson 6.5 l Transformations of Exponential Functions 267

6

Problem 1 Horizontal and Vertical Translations

Earlier you learned that the graph of a function f(x) is translated vertically k units if a

constant k is added to the equation: f(x) � k.

1. Sketch and label the graphs of f(x) � 2x, f(x) � 2x � 3 and f(x) � 2x � 4.

x86

2

4

6

8

–2–2

420–4

–4

–6–8

y

10

12

The graph of a function f(x) is translated horizontally h units if a constant h is

subtracted from the variable in the function: f(x � h).

6.5 TransformersTransformations of Exponential Functions

ObjectiveIn this lesson you will:

l Transform exponential functions algebraically and graphically.

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2. Sketch and label the graphs of f(x) � 2x, f(x) � 2x�3 and f(x) � 2x�4.

3. Sketch and label the graph of f(x) � bx�3 � 5 using the graph of f(x) � bx.

x43

1

2

3

4

–1–1

210–2

–2

–3

–3

–4

–4

y

f( x) = b

x

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Problem 2 ReflectionsEarlier you learned that the graph of a function f(x) is reflected about the x-axis

if the equation of the function is multiplied by negative 1: �f(x).

1. Sketch and label the graph of f(x) � �bx using the graph of f(x) � bx.

x43

1

2

3

4

–1–1

210–2

–2

–3

–3

–4

–4

y

f( x) = bx

Earlier you learned that the graph of a function f(x) is reflected about the y-axis

if the argument (independent variable) of the equation of the function is multiplied

by negative 1: f(�x).

2. Sketch and label the graph of f(x) � b(�x) using the graph of f(x) � bx.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

f( x) = b

x

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3. Sketch and label the graph of f(x) � �b�x using the graph of f(x) � bx.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

f( x) = b

x

Problem 3 DilationsEarlier you learned that the graph of a function f(x) is dilated vertically by a

factor of a if the function is multiplied by a constant a: af(x).

1. Sketch and label the graphs of f(x) � 2x, f(x) � 3(2x), and f(x) � 2 __ 3 (2x).

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The graph of a function f(x) is dilated horizontally by a factor of c if the argument

(independent variable) of the function is multiplied by the constant c: f(cx)

2. Sketch and label the graphs of f(x) � 2x, f(x) � 22x, and f(x) � 2 2 __ 3

x .

3. Rewrite f(x) � 2x�3 and f(x) � 2x�4 in the form of af(x) using the properties

of exponents.

4. What can you conclude about the relationship between a horizontal

translation of h units and a vertical dilation of f(x) � bx based on your results

from Question 3?

5. The vertical dilation of f(x) � abx is identical to the horizontal translation of how

many units?

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Problem 4 Putting it All Together! 1. Sketch and label the graph of f(x) � 2bx � 3 using the graph of f(x) � bx.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

f( x) = b

x

2. Sketch and label the graph of f(x) � �bx�2 � 3 using the graph of f(x) � bx.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

f( x) = b

x

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3. Sketch and label the graph of f(x) � 2b�x�2 using the graph of f(x) � bx.

x86

2

4

6

8

–2–2

420–4

–4

–6

–6

–8

–8

y

f( x) = b

x


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Lesson 6.6 l Exponential Functions and Problem Solving 275

6

Problem 1 Compound InterestThe formula to determine the balance of a compound interest bank account after

t years is

P � P0 ( 1 � r __ n )

nt

where P is the current balance, P0 is the original principal, r is the interest rate

written as a decimal, and n is the number of times per year that the interest

is compounded.

1. Your friend deposited $1000 in a Certificate of Deposit (CD) earning

5.3% interest compounded monthly.

a. Write a function to model the balance of the CD.

b. Graph the function.

x1614

1500

2100

2400

6

900

12104

600

2

300

1200

8

y

1800

6.6 Interest and DecayExponential Functions and Problem Solving

ObjectiveIn this lesson you will:

l Solve problems that are modeled by

exponential functions.

Key Termsl continuous compound

interest formula

l Euler’s number

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c. Graph the linear function y � 2000. Estimate the point of intersection of the

horizontal line and the function.

d. Determine the point of intersection using a graphing calculator.

e. How long will it take the investment to double?

f. How long will it take the investment to be worth $3500?

2. Write an equation to model the balance of an investment of $1000 deposited

in a CD earning 4.5% interest compounded monthly.

3. How long will it take the investment to double?

4. Does the doubling time of an investment depend on the amount invested?

Explain.

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Lesson 6.6 l Exponential Functions and Problem Solving 277

6

The formula for the balance of a bank account after t years in which the interest is

compounded n times per year is P � P0 ( 1 � r __ n )

nt . When the number of times that

the interest is compounded per year increases—from yearly to monthly to weekly to

daily to continuously—the interest is said to compound continuously.

The continuous compound interest formula is a formula for the balance of a

bank account after t years in which the interest is compounded continuously. The

continuous compound interest formula is P � P0ert, where P

0 is the initial principal,

P is the balance after t years, r is the annual interest rate written as a decimal, and

e is Euler’s number, which is approximately equal to 2.71828.

1. One thousand dollars is deposited in a Certificate of Deposit (CD) earning

5.3% interest compounded continuously.

a. Write a function to model the balance of the CD.

b. Graph the function.

x1614

1500

2100

2400

6

900

12104

600

2

300

1200

8

y

1800

c. Graph the linear function y � 2000. Estimate the point of intersection of the

horizontal line and the function.

d. Determine the point of intersection using a graphing calculator.

e. How long will it take for the investment to double?

f. Is the continuous compound interest CD a better investment than the earlier

example of 5.3% interest compounded monthly? How do you know?

Problem 2 Continuous Compound Interest

© 2

010 C

arn

eg

ie L

earn

ing

, In

c.


6

Problem 3 Radioactive DecayThe equation for radioactive decay is N � N

0 2

�t

___ t 1 __ 2

, where N is the amount of

radioactive material after time t when the original amount is N0 and t

1 __ 2 is the half-life.

1. Gold-198 has a half-life of 2.69 days.

a. Write an equation for the decay of 100 kilograms of Gold-198.

b. How much radioactive material is left after one year?

c. How much radioactive material is left after 30 days?

d. Determine when there will be one kilogram left using a graphing calculator.

2. Iron-53 has a half-life of 8.51 minutes.

a. Write an equation for the decay of 50 kilograms of Iron-53.

b. How much radioactive material is left after one day?

c. Determine when there will be one kilogram left using a graphing calculator.


6 exponential functions

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