Polynomial and rational functions are examples of______________functions.

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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.

Victor I.

Numerade Educator

The exponential function $f(x)=e^{x}$ 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________________.

Jesse S.

Numerade Educator

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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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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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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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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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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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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$$Function$$

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

$$Value$$

$$x=24$$

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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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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).]

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

Matthew D.

Numerade Educator

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

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

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

Sydney T.

Numerade Educator

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

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

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

Matita C.

Numerade Educator

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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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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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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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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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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Graphing an Exponential Function In Exercises

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

function.

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

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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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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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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.

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

Phoebe T.

Numerade Educator

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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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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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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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.

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

Jose A.

Numerade Educator

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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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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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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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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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.

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

Justin D.

Numerade Educator

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

exponential function.

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

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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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Using the One-to-One Property In Exercises

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

for $x .$

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

Quiana W.

Numerade Educator

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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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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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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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.

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

Hanan S.

Numerade Educator

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Think About It Which functions are exponential?

(a) 3$x$

(b) 3$x^{2}$

(c) $3^{x}$

(d) $2^{-x}$

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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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