# Precalculus with Limits (2010)

## Educators

LW
KK
+ 2 more educators

### Problem 1

Polynomial and rational functions are examples of _____________ functions.

LW
Luke W.

### 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 given by $f(x)=e^{x}$ 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

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

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ {f(x)=0.9^{x}} & {x=1.4}\end{array}$$

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### Problem 8

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

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=2.3^{x} &x=\frac{3}{2}\end{array}$$

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### Problem 9

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

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=5^{x} & x=-\pi\end{array}$$

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### Problem 10

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

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=\left(\frac{2}{3}\right)^{5 x} & x=\frac{3}{10}\end{array}$$

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### Problem 11

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

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ g(x)=5000\left(2^{x}\right) & x=-1.5\end{array}$$

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### Problem 12

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

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=200(1.2)^{12 x} & x=24\end{array}$$

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### Problem 13

In Exercises $13-16,$ match the exponential function with its graph. [The graphs are labeled (a), (b), ( $(\mathrm{c}),$ and $(\mathrm{d}) . ]$

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

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### Problem 14

In Exercises $13-16,$ match the exponential function with its graph. [The graphs are labeled (a), (b), ( $(\mathrm{c}),$ and $(\mathrm{d}) . ]$

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

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### Problem 15

In Exercises $13-16,$ match the exponential function with its graph. [The graphs are labeled (a), (b), ( $(\mathrm{c}),$ and $(\mathrm{d}) . ]$

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

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### Problem 16

In Exercises $13-16,$ match the exponential function with its graph. [The graphs are labeled (a), (b), ( $(\mathrm{c}),$ and $(\mathrm{d}) . ]$

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

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### Problem 17

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

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

In Exercises $17-22,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

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

KK
Kalpana K.

### Problem 20

In Exercises $17-22,$ use a graphing utility to construct a table of values for the function. Then sketch the graph of the function.

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

Raushan K.

### Problem 21

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

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

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

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

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### Problem 24

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

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

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### Problem 25

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

$$f(x)=2^{x}, g(x)=3-2^{x}$$

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### Problem 26

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

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

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### Problem 27

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

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

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### Problem 28

In Exercises $23-28,$ use the graph of $f$ to describe the transformation that yields the graph of $g .$

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

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### Problem 29

In Exercises $29-32,$ use a graphing utility to graph the exponential function.

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

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### Problem 30

In Exercises $29-32,$ use a graphing utility to graph the exponential function.

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

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### Problem 31

In Exercises $29-32,$ use a graphing utility to graph the exponential function.

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

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### Problem 32

In Exercises $29-32,$ use a graphing utility to graph the exponential function.

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

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### Problem 33

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ h(x)=e^{-x} & x=\frac{3}{4}\end{array}$$

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### Problem 34

In Exercises $33-38$ , 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 35

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=2 e^{-5 x} & x=10\end{array}$$

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### Problem 36

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=1.5 e^{x / 2} & x=240\end{array}$$

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### Problem 37

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=5000 e^{0.06 x} & x=6\end{array}$$

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### Problem 38

In Exercises $33-38$ , evaluate the function at the indicated value of $x$ . Round your result to three decimal places.

$$\begin{array}{ll}{\text { Function }} & {\text { Value }} \\ f(x)=250 e^{0.05 x} & x=20\end{array}$$

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### Problem 39

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

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

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

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

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

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

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

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

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### Problem 46

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

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

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### Problem 47

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

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

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

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

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### Problem 50

In Exercises $45-50$ , use a graphing utility to graph the exponential function.

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

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### Problem 51

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

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

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### Problem 52

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

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

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### Problem 53

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

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

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### Problem 54

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

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

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### Problem 55

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

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

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### Problem 56

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

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

Charles S.

### Problem 57

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

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

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### Problem 58

In Exercises $51-58$ , use the One-to-One Property to solve the equation for $x .$

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

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### Problem 59

In Exercises $59-62,$ complete the table to determine the balance $A$ for $P$ dollars invested at
rate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous }} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

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

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### Problem 60

In Exercises $59-62,$ complete the table to determine the balance $A$ for $P$ dollars invested atrate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous }} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

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

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### Problem 61

In Exercises $59-62,$ complete the table to determine the balance $A$ for $P$ dollars invested atrate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous }} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

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

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### Problem 62

In Exercises $59-62,$ complete the table to determine the balance $A$ for $P$ dollars invested atrate $r$ for $t$ years and compounded $n$ times per year.
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {4} & {12} & {365} & {\text { Continuous }} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

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

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In Exercises $63-66,$ complete the table to determine the balance $A$ for $\$ 12,000$invested at rate$r$for$t$years, compounded continuously. $$\begin{array}{|c|c|c|c|c|c|}\hline t & {10} & {20} & {30} & {40} & {50} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$ $$r=4 \%$$ Check back soon! ### Problem 64 In Exercises$63-66,$complete the table to determine the balance$A$for$\$12,000$ invested at rate $r$ for $t$ years, compounded continuously.

$$\begin{array}{|c|c|c|c|c|c|}\hline t & {10} & {20} & {30} & {40} & {50} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$r=6 \%$$

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In Exercises $63-66,$ complete the table to determine the balance $A$ for $\$ 12,000$invested at rate$r$for$t$years, compounded continuously. $$\begin{array}{|c|c|c|c|c|c|}\hline t & {10} & {20} & {30} & {40} & {50} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$ $$r=6.5 \%$$ Check back soon! ### Problem 66 In Exercises$63-66,$complete the table to determine the balance$A$for$\$12,000$ invested at rate $r$ for $t$ years, compounded continuously.

$$\begin{array}{|c|c|c|c|c|c|}\hline t & {10} & {20} & {30} & {40} & {50} \\ \hline A & {} & {} & {} & {} & {} \\ \hline\end{array}$$

$$r=3.5 \%$$

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### Problem 69

If 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. Darin B. Numerade Educator ### Problem 70 The number$V$of computers infected by a computer virus increases according to the model$V(t)=100 e^{4.4602 t}$, where$t$is the time in hours. Find the number of computers infected after (a) 1 hour, (b) 1.5 hours, and (c) 2 hours. Check back soon! ### Problem 71 The projected populations of California for the years 2015 through 2030 can be modeled by$P=34.696 e^{0.0098 t},$where$P$is the population (in millions) and$t$is the time (in years), with$t=15$corresponding to 2015 . (Source: U.S. Census Bureau) (a) Use a graphing utility to graph the function for the years 2015 through 2030 . (b) Use the table feature of a graphing utility to create a table of values for the same time period as in part (a). (c) According to the model, when will the population of California exceed 50 million? Check back soon! ### Problem 72 The populations$P$(in millions) of Italy from 1990 through 2008 can be approximated by the model$P=56.8 e^{0.0015 t},$where$t$represents the year, with$t=0$corresponding to$1990 .$(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 2008 . (c) Use the model to predict the populations of Italy in 2015 and 2020 . Gregory S. Numerade Educator ### Problem 73 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$t=150,000$. Check back soon! ### Problem 74 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$t=10,00$. Deboney H. Numerade Educator ### Problem 75 After$t$years, the value of a wheel-chair conversion van that originally cost$\$30,500$ 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
(b) Determine the value of the van 4 years after it was purchased.

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### Problem 76

Immediately following an injection, the concentration of a drug in the bloodstream is 300 milligrams per milliliter. After $t$ hours, the 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 77

TRUE OR FALSE? In Exercises 77 and $78,$ determine whether 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 78

TRUE OR FALSE? In Exercises 77 and $78,$ determine whether the statement is true or false. Justify your answer.

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

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### Problem 79

THINK ABOUT IT In Exercises $79-82,$ 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 80

THINK ABOUT IT In Exercises $79-82,$ 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 81

In Exercises $79-82,$ use properties of exponents to determine which functions (if any) are the same.

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

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### Problem 82

In Exercises $79-82,$ 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 83

Graph the functions given by $y=3^{x}$ and $y=4^{x}$ and use the graphs to solve each inequality.
$\begin{array}{ll}{\text { (a) } 4^{x}<3^{x}} & {\text { (b) } 4^{x}>3^{x}}\end{array}$

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### Problem 84

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.
$\begin{array}{ll}{\text { (a) } f(x)=x^{2} e^{-x}} & {\text { (b) } g(x)=x 2^{3-x}}\end{array}$

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### Problem 85

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 86

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 87

Use a graphing utility to graph each pair of functions in the same viewing window. Describe any similarities and differences in the graphs.
$\begin{array}{lll}{\text { (a) } y_{1}=2^{x}, y_{2}=x^{2}} & {\text { (b) } y_{1}=3^{x}, y_{2}=x^{3}}\end{array}$

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### Problem 88

THINK ABOUT IT Which functions are exponential?
$\begin{array}{llll}{\text { (a) } 3 x} & {\text { (b) } 3 x^{2}} & {\text { (c) } 3^{x}} & {\text { (d) } 2^{-x}}\end{array}$

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### Problem 89

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

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