Polynomial and rational functions are examples of _____________ functions.

Luke W.

Numerade Educator

Exponential and logarithmic functions are examples of nonalgebraic functions, also called ____________ functions.

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You can use the ______________ Property to solve simple exponential equations.

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

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

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

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

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

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

Kalpana K.

Numerade Educator

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

Raushan K.

Numerade Educator

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Charles S.

Numerade Educator

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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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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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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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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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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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.

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

$$r=4 \%$$

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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.

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

$$r=6 \%$$

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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.

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

$$r=6.5 \%$$

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$$r=3.5 \%$$

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On the day of a child's birth, a deposit of $\$ 30,000$ is made in a trust fund that pays 5$\%$ interest, compounded continuously. Determine the balance in this account on the child's 25 th birthday.

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A deposit of $\$ 5000$ is made in a trust fund that pays 7.5$\%$ interest, compounded continuously. It is specified that the balance will be given to the college from which the donor graduated after the money has earned interest for 50 years. How much will the college receive?

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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

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.

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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?

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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

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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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

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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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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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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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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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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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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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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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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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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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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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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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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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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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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Use the formula

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

to calculate the balance of an account 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 of the account? Explain.

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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. [The graphs are labeled (a) through (f).] Explain your reasoning.

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