Problem 1

Polynomial and rational functions are examples of______________functions.

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Problem 2

Exponential and logarithmic functions are examples of nonalgebraic functions, also called____________functions.

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Problem 3

You can use the___________ Property to solve simple exponential equations.

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Problem 4

The exponential function $f(x)=e^{x}$ is called the____________ _____________function, and the base $e$ is called the____________base.

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Problem 5

To find the amount $A$ in an account after $t$ years with principal $P$ and an annual interest rate $r$ compounded $n$ times per year, you can use the formula________________.

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Problem 6

To find the amount $A$ in an account after $t$ years with principal $P$ and an annual interest rate $r$ compounded continuously, you can use the formula______________.

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Problem 7

Evaluating an Exponential Function In Exercises

$7-12$ , evaluate the function at the indicated value of $x .$

Round your result to three decimal places.

Function Value

$f(x)=0.9^{x}$ $x=1.4$

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Problem 8

Evaluating an Exponential Function In Exercises

$7-12$ , evaluate the function at the indicated value of $x .$

Round your result to three decimal places.

$$Function$$

$$f(x)=0.9^{x}$$

$$Value$$

$$x=1.4$$

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Problem 9

Evaluating an Exponential Function In Exercises $7-12,$ evaluate the function at the indicated value of $x .$ Round your result to three decimal places.

$$Function$$

$$f(x)=5^{x}$$

$$Value$$

$$x=-\pi$$

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Problem 10

Evaluating an Exponential Function In Exercises $7-12,$ evaluate the function at the indicated value of $x .$ Round your result to three decimal places.

$$Function$$

$$f(x)=\left(\frac{2}{3}\right)^{5 x}$$

$$Value$$

$$x=\frac{3}{10}$$

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Problem 11

Evaluating an Exponential Function In Exercises $7-12,$ evaluate the function at the indicated value of $x .$ Round your result to three decimal places.

$$Function$$

$$g(x)=5000\left(2^{x}\right)$$

$$Value$$

$$x=-1.5$$

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Problem 12

$$Function$$

$$f(x)=200(1.2)^{12 x}$$

$$Value$$

$$x=24$$

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Problem 13

Matching an Exponential Function with Its

Graph In Exercises $13-16,$ match the exponential

function with its graph. [The graphs are labeled (a), (b),

(c), and (d).]

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Problem 13

Matching an Exponential Function with Its

Graph In Exercises $13-16,$ match the exponential

function with its graph. [The graphs are labeled (a), (b),

(c), and (d).]

$$f(x)=2^{x}$$

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Problem 14

Matching an Exponential Function with Its

Graph In Exercises $13-16,$ match the exponential

function with its graph. [The graphs are labeled (a), (b),

(c), and (d).]

$$f(x)=2^{x}+1$$

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Problem 15

Matching an Exponential Function with Its

Graph In Exercises $13-16,$ match the exponential

function with its graph. [The graphs are labeled (a), (b),

(c), and (d).]

$$f(x)=2^{-x}$$

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Problem 16

Matching an Exponential Function with Its

Graph In Exercises $13-16,$ match the exponential

function with its graph. [The graphs are labeled (a), (b),

(c), and (d).]

$$f(x)=2^{x-2}$$

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Problem 17

Graphing an Exponential Function In Exercises

$17-22,$ use a graphing utility to construct a table of values

for the function. Then sketch the graph of the function.

$$f(x)=\left(\frac{1}{2}\right)^{x}$$

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Problem 18

Graphing an Exponential Function In Exercises

$17-22,$ use a graphing utility to construct a table of values

for the function. Then sketch the graph of the function.

$$f(x)=\left(\frac{1}{2}\right)^{-x}$$

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Problem 19

Graphing an Exponential Function In Exercises

$17-22,$ use a graphing utility to construct a table of values

for the function. Then sketch the graph of the function.

$$f(x)=6^{-x}$$

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Problem 20

Graphing an Exponential Function In Exercises

$17-22,$ use a graphing utility to construct a table of values

for the function. Then sketch the graph of the function.

$$f(x)=6^{x}$$

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Problem 21

Graphing an Exponential Function In Exercises

$17-22,$ use a graphing utility to construct a table of values

for the function. Then sketch the graph of the function.

$$f(x)=2^{x-1}$$

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Problem 22

Graphing an Exponential Function In Exercises

$17-22,$ use a graphing utility to construct a table of values

for the function. Then sketch the graph of the function.

$$f(x)=4^{x-3}+3$$

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Problem 23

Using the One-to-One Property In Exercises $23-26$

use the One-to-One Property to solve the equation for $x .$

$$3^{x+1}=27$$

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Problem 24

Using the One-to-One Property In Exercises $23-26$

use the One-to-One Property to solve the equation for $x .$

$$2^{x-3}=16$$

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Problem 25

Using the One-to-One Property In Exercises $23-26$

use the One-to-One Property to solve the equation for $x .$

$$\left(\frac{1}{2}\right)^{x}=32$$

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Problem 26

Using the One-to-One Property In Exercises $23-26$

use the One-to-One Property to solve the equation for $x .$

$$5^{x-2}=\frac{1}{125}$$

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Problem 27

Transforming the Graph of an Exponential

Function In Exercises $27-30,$ use the graph of $f$ to

describe the transformation that yields the graph of $g$ .

$$f(x)=3^{x}, \quad g(x)=3^{x}+1$$

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Problem 28

Transforming the Graph of an Exponential

Function In Exercises $27-30,$ use the graph of $f$ to

describe the transformation that yields the graph of $g$ .

$$f(x)=10^{x}, \quad g(x)=10^{-x+3}$$

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Problem 29

Transforming the Graph of an Exponential

Function In Exercises $27-30,$ use the graph of $f$ to

describe the transformation that yields the graph of $g$ .

$$f(x)=\left(\frac{7}{2}\right)^{x}, \quad g(x)=-\left(\frac{7}{2}\right)^{-x}$$

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Problem 30

Transforming the Graph of an Exponential

Function In Exercises $27-30,$ use the graph of $f$ to

describe the transformation that yields the graph of $g$ .

$$f(x)=0.3^{x}, \quad g(x)=-0.3^{x}+5$$

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Problem 31

Graphing an Exponential Function In Exercises

$31-34,$ use a graphing utility to graph the exponential

function.

$$y=2^{-x^{2}}$$

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Problem 32

Graphing an Exponential Function In Exercises

$31-34,$ use a graphing utility to graph the exponential

function.

$$y=3^{-|x|}$$

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Problem 33

Graphing an Exponential Function In Exercises

$31-34,$ use a graphing utility to graph the exponential

function.

$$y=3^{x-2}+1$$

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Problem 34

Graphing an Exponential Function In Exercises

$31-34,$ use a graphing utility to graph the exponential

function.

$$y=4^{x+1}-2$$

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Problem 35

Evaluating a Natural Exponential Function In

Exercises $35-38$ , evaluate the function at the indicated

value of $x .$ Round your result to three decimal places.

$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ {f(x)=e^{x}} & {x=3.2}\end{array}$

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Problem 36

Evaluating a Natural Exponential Function In

Exercises $35-38$ , evaluate the function at the indicated

value of $x .$ Round your result to three decimal places.

$$functions$$

$$f(x)=1.5 e^{x / 2}$$

$$values$$

$$x=240$$

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Problem 37

Evaluating a Natural Exponential Function In

Exercises $35-38$ , evaluate the function at the indicated

value of $x .$ Round your result to three decimal places.

$$functions$$

$$f(x)=5000 e^{0.06 x}$$

$$values$$

$$x=6$$

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Problem 38

Evaluating a Natural Exponential Function In

Exercises $35-38$ , evaluate the function at the indicated

value of $x .$ Round your result to three decimal places.

$$functions$$

$$f(x)=250 e^{0.05 x}$$

$$values$$

$$x=20$$

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Problem 39

Graphing a Natural Exponential Function In

Exercises $39-44$ , use a graphing utility to construct a

table of values for the function. Then sketch the graph of

the function.

$$f(x)=e^{x}$$

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Problem 40

Graphing a Natural Exponential Function In

Exercises $39-44$ , use a graphing utility to construct a

table of values for the function. Then sketch the graph of

the function.

$$f(x)=e^{-x}$$

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Problem 41

Graphing a Natural Exponential Function In

Exercises $39-44$ , use a graphing utility to construct a

table of values for the function. Then sketch the graph of

the function.

$$f(x)=3 e^{x+4}$$

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Problem 42

Graphing a Natural Exponential Function In

Exercises $39-44$ , use a graphing utility to construct a

table of values for the function. Then sketch the graph of

the function.

$$f(x)=2 e^{-0.5 x}$$

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Problem 43

Graphing a Natural Exponential Function In

Exercises $39-44$ , use a graphing utility to construct a

table of values for the function. Then sketch the graph of

the function.

$$f(x)=2 e^{x-2}+4$$

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Problem 44

Graphing a Natural Exponential Function In

Exercises $39-44$ , use a graphing utility to construct a

table of values for the function. Then sketch the graph of

the function.

$$f(x)=2+e^{x-5}$$

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Problem 45

Graphing a Natural Exponential Function In

Exercises $45-50$ , use a graphing utility to graph the

exponential function.

$$y=1.08 e^{-5 x}$$

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Problem 46

Graphing a Natural Exponential Function In

Exercises $45-50$ , use a graphing utility to graph the

exponential function.

$$y=1.08 e^{5 x}$$

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Problem 47

Graphing a Natural Exponential Function In

Exercises $45-50$ , use a graphing utility to graph the

exponential function.

$$s(t)=2 e^{0.12 t}$$

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Problem 48

Graphing a Natural Exponential Function In

Exercises $45-50$ , use a graphing utility to graph the

exponential function.

$$s(t)=3 e^{-0.2 t}$$

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Problem 49

Graphing a Natural Exponential Function In

Exercises $45-50$ , use a graphing utility to graph the

exponential function.

$$g(x)=1+e^{-x}$$

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Problem 49

Exercises $45-50$ , use a graphing utility to graph the

exponential function.

$$g(x)=1+e^{-x}$$

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Problem 50

Graphing a Natural Exponential Function In

Exercises $45-50$ , use a graphing utility to graph the

exponential function.

$$h(x)=e^{x-2}$$

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Problem 51

Using the One-to-One Property In Exercises

$51-54,$ use the One-to-One Property to solve the equation

for $x .$

$$e^{3 x+2}=e^{3}$$

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Problem 52

Using the One-to-One Property In Exercises

$51-54,$ use the One-to-One Property to solve the equation

for $x .$

$$e^{2 x-1}=e^{4}$$

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Problem 53

Using the One-to-One Property In Exercises

$51-54,$ use the One-to-One Property to solve the equation

for $x .$

$$e^{x^{2}-3}=e^{2 x}$$

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Problem 54

Using the One-to-One Property In Exercises

$51-54,$ use the One-to-One Property to solve the equation

for $x .$

$$e^{x^{2}+6}=e^{5 x}$$

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Problem 55

Compound Interest In Exercises $55-58$ , complete

the table to determine the balance $A$ for $P$ dollars invested

at rate $r$ for $t$ years and compounded $n$ times per year.

$$P=\$ 1500, r=2 \%, t=10$$

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Problem 56

Compound Interest In Exercises $55-58$ , complete

the table to determine the balance $A$ for $P$ dollars invested

at rate $r$ for $t$ years and compounded $n$ times per year.

$$P=\$ 2500, r=3.5 \%, t=10$$

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Problem 57

Compound Interest In Exercises $55-58$ , complete

the table to determine the balance $A$ for $P$ dollars invested

at rate $r$ for $t$ years and compounded $n$ times per year.

$$P=\$ 2500, r=4 \%, t=20$$

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Problem 58

Compound Interest In Exercises $55-58$ , complete

the table to determine the balance $A$ for $P$ dollars invested

at rate $r$ for $t$ years and compounded $n$ times per year.

$$P=\$ 1000, r=6 \%, t=40$$

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Problem 59

Compound Interest In Exercises $59-62,$ complete

the table to determine the balance $A$ for $\$ 12,000$ invested

at rate $r$ for $t$ years, compounded continuously.

$$r=4 \%$$

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Problem 60

Compound Interest In Exercises $59-62,$ complete

the table to determine the balance $A$ for $\$ 12,000$ invested

at rate $r$ for $t$ years, compounded continuously.

$$r=6 \%$$

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Problem 61

Compound Interest In Exercises $59-62,$ complete

the table to determine the balance $A$ for $\$ 12,000$ invested

at rate $r$ for $t$ years, compounded continuously.

$$r=6.5 \%$$

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Problem 62

Compound Interest In Exercises $59-62,$ complete

the table to determine the balance $A$ for $\$ 12,000$ invested

at rate $r$ for $t$ years, compounded continuously.

$$r=3.5 \%$$

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Problem 63

Trust Fund On the day of a child's birth, a parest,

deposits $\$ 30,000$ in a trust fund that pays 5$\%$ interest,

compounded continuouly. Determine the balance in

this account on the child's 25 th birthday.

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Problem 64

Trust Fund A philanthropist deposits $\$ 5000$ in

a trust fund that pays 7.5$\%$ interest, compounded

continuously. The balance will be given to the college

from which the philance will be given to the college

money has earned interest for 50 years. How much will

the college receive?

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Problem 65

Inflation Assuming that the annual rate of inflation

averages 4$\%$ over the next 10 years, the approximate

costs $C$ of goods or services during any year in that

decade will be modeled by $C(t)=P(1.04)^{t},$ where $t$ is

the time in years and $P$ is the present cost. The price of

an oil change for your car is presently $\$ 23.95 .$ Estimate

the price 10 years from now.

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Problem 66

Computer Virus The number $V$ of computers

infected by a virus increases according to the model

$V(t)=100 e^{4.6052 t},$ where $t$ is the time in hours. Find

the number of computers infected after (a) 1 hour,

(b) 1.5 hours, and $(\mathrm{c}) 2$ hours.

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Problem 67

Population Growth The projected populations

of the United States for the years 2020 through 2050

can be modeled by $P=290.323 e^{0.0083 t}$ , where $P$ is the

population (in millions) and $t$ is the time (in years), with

$t=20$ corresponding to $2020 . \quad$ (Source: U.S. Census

Bureau)

(a) Use a graphing utility to graph the function for the

years 2020 through 2050 .

(b) Use the table feature of the graphing utility to

create a table of values for the same time period as

in part (a).

(c) According to the model, during what year will the

population of the United States exceed 400 million?

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Problem 68

Population The populations $P$ (in millions) of Italy

from 2000 through 2012 can be approximated by the

model $P=57.563 e^{0.0052}$ , where $t$ represents the year,

with $t=0$ corresponding to $2000 .$ (Source: U.S.

Census Bureau, International Data Base)

(a) According to the model, is the population of Italy

increasing or decreasing? Explain.

(b) Find the populations of Italy in 2000 and 2012 .

(c) Use the model to predict the populations of Italy in

2020 and $2025 .$

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Problem 69

Radioactive Decay Let $Q$ represent a mass of

radioactive plutonium $(239 \mathrm{Pu})($ in grams), whose half-life

is $24,100$ years. The quantity of plutonium present after

$t$ years is $Q=16\left(\frac{1}{2}\right)^{t / 24,100}$

(a) Determine the initial quantity (when $t=0 )$ .

(b) Determine the quantity present after $75,000$ years.

(c) Use a graphing utility to graph the function over the

interval $t=0$ to $t=150,000 .$

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Problem 70

Radioactive Decay Let $Q$ represent a mass of

carbon 14$(14 \mathrm{C})$ (in grams), whose half-life is 5715 years.

The quantity of carbon 14 present after $t$ years is

$Q=10\left(\frac{1}{2}\right)^{t / 5715}$ .

(a) Determine the initial quantity (when $t=0 )$ .

(b) Determine the quantity present after 2000 years.

(c) Sketch the graph of this function over the interval

$t=0$ to $t=10,000 .$

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Problem 71

Depreciation After $t$ years, the value of a wheelchair

conversion van that originally cost $\$ 49,810$ depreciates so

that each year it is worth $\frac{7}{8}$ of its value for the previous year.

(a) Find a model for $V(t),$ the value of the van after

$t$ years.

(b) Determine the value of the van 4 years after it was

purchased.

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Problem 72

Immediately following an injection, the concentration of a drug in the

bloodstream is

300 milligrams

per milligrams

After $t$ hours, the

concentration is

75$\%$ of the level of

the previous hour.

(a) Find a model for

$C(t)$ , the concentration of the drug after $t$ hours.

(b) Determine the concentration of the drug after

8 hours.

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Problem 73

True or False? In Exercises 73 and $74,$ determinewhether the statement is true or false. Justify your

answer.

The line $y=-2$ is an asymptote for the graph of

$f(x)=10^{x}-2$

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Problem 74

True or False? In Exercises 73 and $74,$ determine whether the statement is true or false. Justify your

answer.

$$e=\frac{271,801}{99,990}$$

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Problem 75

Think About It In Exercises $75-78$ , use properties of

exponents to determine which functions (if any) are the

same.

$$\begin{array}{l}{f(x)=3^{x-2}} \\ {g(x)=3^{x}-9} \\ {h(x)=\frac{1}{9}\left(3^{x}\right)}\end{array}$$

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Problem 76

Think About It In Exercises $75-78$ , use properties of

exponents to determine which functions (if any) are the

same.

$$\begin{array}{l}{f(x)=4^{x}+12} \\ {g(x)=2^{2 x+6}} \\ {h(x)=64\left(4^{x}\right)}\end{array}$$

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Problem 77

Think About It In Exercises $75-78$ , use properties of

exponents to determine which functions (if any) are the

same.

$$\begin{aligned} f(x) &=16\left(4^{-x}\right) \\ g(x) &=\left(\frac{1}{4}\right)^{x-2} \\ h(x) &=16\left(2^{-2 x}\right) \end{aligned}$$

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Problem 78

Think About It In Exercises $75-78$ , use properties of

exponents to determine which functions (if any) are the

same.

$$\begin{array}{l}{f(x)=e^{-x}+3} \\ {g(x)=e^{3-x}} \\ {h(x)=-e^{x-3}}\end{array}$$

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Problem 79

Solving Inequalities Graph the functions $y=3^{x}$

and $y=4^{x}$ and use the graphs to solve each inequality.

(a) $4^{x}<3^{x}$

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Problem 80

Graphical Analysis Use a graphing utility to graph

each function. Use the graph to find where the function

is increasing and decreasing, and approximate any

relative maximum or minimum values.

(a) $f(x)=x^{2} e^{-x} \quad$ (b) $g(x)=x 2^{3-x}$

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Problem 81

Graphical Analysis Use a graphing utility to

graph $y_{1}=(1+1 / x)^{x}$ and $y_{2}=e$ in the same viewing

window. Using the trace feature, explain what happens

to the graph of $y_{1}$ as $x$ increases.

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Problem 82

Graphical Analysis Use a graphing utility to graph

$f(x)=\left(1+\frac{0.5}{x}\right)^{x}$ and $g(x)=e^{0.5}$

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Problem 82

Graphical Analysis Use a graphing utility to graph

$f(x)=\left(1+\frac{0.5}{x}\right)^{x}$ and $g(x)=e^{0.5}$

in the same viewing window. What is the relationship

between $f$ and $g$ as $x$ increases and decreases without

bound?

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Problem 83

Graphical Analysis Use a graphing utility to graph

each pair of functions in the same viewing window.

Describe any similarities and differences in the graphs.

(a) $y_{1}=2^{x}, y_{2}=x^{2}$

(b) $y_{1}=3^{x}, y_{2}=x^{3}$

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Problem 84

HOW DO YOU SEE IT? The figure shows

the graphs of $y=2^{x}, y=e^{x}, y=10^{x}$ ,

$y=2^{-x}, y=e^{-x}$ , and $y=10^{-x}$ . Match each

function with its graph. IThe graphs are labeled

(a) through (f). Explain your reasoning.

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Problem 85

Think About It Which functions are exponential?

(a) 3$x$

(b) 3$x^{2}$

(c) $3^{x}$

(d) $2^{-x}$

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Problem 86

Compound Interest Use the formula

$A=P\left(1+\frac{r}{n}\right)^{n t}$

to calculate the balance of an investment when $P=\$ 3000$ ,

$r=6 \%,$ and $t=10$ years, and compounding is done

(a) by the day, (b) by the hour, (c) by the minute, and

(d) by the second. Does increasing the number of

compoundings per year result in unlimited growth of

the balance? Explain.

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