If $x=-2,$ the value of the expression $3 x^{2}-5 x+\frac{1}{x}$ is _________.

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The domain of the variable in the expression $\frac{x-3}{x+4}$ is ________.

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To rationalize the denominator of $\frac{3}{\sqrt{5}-2},$ multiply the numerator and denominator by _______.

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For a function $y=f(x),$ the variable $x$ is the _______ variable, and the variable $y$ is the ____ variable.

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Multiple Choice The set of all images of the elements in the domain of a function is called the ____.

(a) range

(b) domain

(c) solution set

(d) function

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Multiple Choice The independent variable is sometimes referred to as the ______ of the function.

(a) range

(b) value

(c) argument

(d) definition

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True or False The domain of $\frac{f}{g}$ consists of the numbers $x$ that are in the domains of both $f$ and $g$.

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Four ways of expressing a relation are _______, ________, _________, and __________.

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True or False If no domain is specified for a function $f$, then the domain of $f$ is the set of real numbers.

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True or False If $x$ is in the domain of a function $f,$ we say that $f$ is not defined at $x,$ or $f(x)$ does not exist.

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In Problems 17 and 18 , a relation expressed verbally is given.

(a) What is the domain and the range of the relation?

(b) Express the relation using a mapping.

(c) Express the relation as a set of ordered pairs.

The density of a gas under constant pressure depends on temperature. Holding pressure constant at 14.5 pounds per square inch, a chemist measures the density of an oxygen sample at temperatures of $0,22,40,70,$ and $100^{\circ} \mathrm{C}$ and obtains densities of $1.411,1.305,1.229,1.121,$ and $1.031 \mathrm{~kg} / \mathrm{m}^{3},$ respectively.

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A relation expressed verbally is given.

(a) What is the domain and the range of the relation?

(b) Express the relation using a mapping.

(c) Express the relation as a set of ordered pairs.

A researcher wants to investigate how weight depends on height among adult males in Europe. She visits five regions in Europe and determines the average heights in those regions to be $1.80,1.78,1.77,1.77,$ and 1.80 meters. The corresponding average weights are $87.1,86.9,83.0,84.1,$ and $86.4 \mathrm{~kg},$ respectively.

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In Problems 19-30, find the domain and range of each relation. Then determine whether the relation represents a function.

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Find the domain and range of each relation. Then determine whether the relation represents a function.

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Find the domain and range of each relation. Then determine whether the relation represents a function.

$$

\{(2,6),(-3,6),(4,9),(2,10)\}

$$

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Find the domain and range of each relation. Then determine whether the relation represents a function.

$$

\{(-2,5),(-1,3),(3,7),(4,12)\}

$$

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Find the domain and range of each relation. Then determine whether the relation represents a function.

$$

\{(1,3),(2,3),(3,3),(4,3)\}

$$

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Find the domain and range of each relation. Then determine whether the relation represents a function.

$$

\{(0,-2),(1,3),(2,3),(3,7)\}

$$

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Find the domain and range of each relation. Then determine whether the relation represents a function.

$$

\{(3,3),(3,5),(0,1),(-4,6)\}

$$

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Find the domain and range of each relation. Then determine whether the relation represents a function.

$$

\{(-4,4),(-3,3),(-2,2),(-1,1),(-4,0)\}

$$

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Find the domain and range of each relation. Then determine whether the relation represents a function.

$$

\{(-1,8),(0,3),(2,-1),(4,3)\}

$$

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Find the domain and range of each relation. Then determine whether the relation represents a function.

$$

\{(-2,16),(-1,4),(0,3),(1,4)\}

$$

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In Problems $31-42,$ determine whether the equation defines $y$ as a function of $x .$

$$

y=2 x^{2}-3 x+4

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

y=x^{3}

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

y=\frac{1}{x}

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

y=|x|

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

x^{2}=8-y^{2}

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

y=\pm \sqrt{1-2 x}

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

x=y^{2}

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

x+y^{2}=1

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

y=\sqrt[3]{x}

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

y=\frac{3 x-1}{x+2}

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

|y|=2 x+3

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

|y|=2 x+3

$$

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Determine whether the equation defines $y$ as a function of $x .$

$$

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

$$

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In Problems 43-50, find the following for each function:

(a) $f(0)$

(b) $f(1)$

(c) $f(-1)$

(d) $f(-x)$

(e) $-f(x)$

(f) $f(x+1)$

(g) $f(2 x)$

(h) $f(x+h)$

$$

f(x)=3 x^{2}+2 x-4

$$

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Find the following for each function:

(a) $f(0)$

(b) $f(1)$

(c) $f(-1)$

(d) $f(-x)$

(e) $-f(x)$

(f) $f(x+1)$

(g) $f(2 x)$

(h) $f(x+h)$

$$

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

$$

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Find the following for each function:

(a) $f(0)$

(b) $f(1)$

(c) $f(-1)$

(d) $f(-x)$

(e) $-f(x)$

(f) $f(x+1)$

(g) $f(2 x)$

(h) $f(x+h)$

$$

f(x)=\frac{x}{x^{2}+1}

$$

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Find the following for each function:

(a) $f(0)$

(b) $f(1)$

(c) $f(-1)$

(d) $f(-x)$

(e) $-f(x)$

(f) $f(x+1)$

(g) $f(2 x)$

(h) $f(x+h)$

$$

f(x)=\frac{x^{2}-1}{x+4}

$$

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Find the following for each function:

(a) $f(0)$

(b) $f(1)$

(c) $f(-1)$

(d) $f(-x)$

(e) $-f(x)$

(f) $f(x+1)$

(g) $f(2 x)$

(h) $f(x+h)$

$$

f(x)=|x|+4

$$

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Find the following for each function:

(a) $f(0)$

(b) $f(1)$

(c) $f(-1)$

(d) $f(-x)$

(e) $-f(x)$

(f) $f(x+1)$

(g) $f(2 x)$

(h) $f(x+h)$

$$

f(x)=\sqrt{x^{2}+x}

$$

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Find the following for each function:

(a) $f(0)$

(b) $f(1)$

(c) $f(-1)$

(d) $f(-x)$

(e) $-f(x)$

(f) $f(x+1)$

(g) $f(2 x)$

(h) $f(x+h)$

$$

f(x)=\frac{2 x+1}{3 x-5}

$$

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Find the following for each function:

(a) $f(0)$

(b) $f(1)$

(c) $f(-1)$

(d) $f(-x)$

(e) $-f(x)$

(f) $f(x+1)$

(g) $f(2 x)$

(h) $f(x+h)$

$$

f(x)=1-\frac{1}{(x+2)^{2}}

$$

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Find the domain of each function.

$$

M(t)=\sqrt[5]{\frac{t+1}{t^{2}-5 t-14}}

$$

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Find the domain of each function.

$$

N(p)=\sqrt[5]{\frac{p}{2 p^{2}-98}}

$$

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In Problems $71-80$, for the given functions fand g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=3 x+4 ; \quad g(x)=2 x-3

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=2 x+1 ; \quad g(x)=3 x-2

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=x-1 ; \quad g(x)=2 x^{2}

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=2 x^{2}+3 ; \quad g(x)=4 x^{3}+1

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=\sqrt{x} ; \quad g(x)=3 x-5

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=1+\frac{1}{x} ; \quad g(x)=\frac{1}{x}

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=\sqrt{x-1} ; \quad g(x)=\sqrt{4-x}

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=\frac{2 x+3}{3 x-2} ; \quad g(x)=\frac{4 x}{3 x-2}

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

f(x)=\sqrt{x+1} ; \quad g(x)=\frac{2}{x}

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

\text { Given } f(x)=3 x+1 \text { and }(f+g)(x)=6-\frac{1}{2} x, \text { find the function } g

$$

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For the given functions f and g, find the following. For parts $(a)-(d)$, also find the domain.

$$

\text { Given } f(x)=\frac{1}{x} \text { and }\left(\frac{f}{g}\right)(x)=\frac{x+1}{x^{2}-x}, \text { find the function } g

$$

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In Problems $83-98$, find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=4 x+3

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=-3 x+1

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=x^{2}-4

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=3 x^{2}+2

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

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

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=3 x^{2}-2 x+6

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\frac{5}{4 x-3}

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\frac{1}{x+3}

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\frac{2 x}{x+3}

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\frac{5 x}{x-4}

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\sqrt{x-2}

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\sqrt{x+1}

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

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

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\frac{1}{x^{2}+1}

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\sqrt{4-x^{2}}

$$

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Find the difference quotient of $f$; that is, find $\frac{f(x+h)-f(x)}{h}, h \neq 0,$ for each function. Be sure to simplify.

$$

f(x)=\frac{1}{\sqrt{x+2}}

$$

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If $f(x)=\frac{5}{6} x-\frac{3}{4},$ find the value $(\mathrm{s})$ of $x$ so that $f(x)=-\frac{7}{16}$

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If $f(x)=2 x^{3}+A x^{2}+4 x-5$ and $f(2)=5,$ what is the value of $A ?$

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If $f(x)=\frac{2 x-B}{3 x+4}$ and $f(2)=\frac{1}{2},$ what is the value of $B ?$

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Geometry Express the area $A$ of a rectangle as a function of the length $x$ if the length of the rectangle is twice its width.

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Geometry Express the area $A$ of an isosceles right triangle as a function of the length $x$ of one of the two equal sides.

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Constructing Functions Express the gross wages $G$ of a person who earns $\$ 16$ per hour as a function of the number $x$ of hours worked.

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Constructing Functions Ann, a commissioned salesperson, earns $\$ 100$ base pay plus $\$ 10$ per item sold. Express her gross salary $G$ as a function of the number $x$ of items sold.

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Effect of Gravity on Earth If a rock falls from a height of 20 meters on Earth, the height $H$ (in meters) after $x$ seconds is approximately

$$

H(x)=20-4.9 x^{2}

$$

(a) What is the height of the rock when $x=1$ second? When $x=1.1$ seconds? When $x=1.2$ seconds?

(b) When is the height of the rock 15 meters? When is it 10 meters? When is it 5 meters?

(c) When does the rock strike the ground?

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Effect of Gravity on Jupiter If a rock falls from a height of 20 meters on the planet Jupiter, its height $H$ (in meters) after $x$ seconds is approximately

$$

H(x)=20-13 x^{2}

$$

(a) What is the height of the rock when $x=1$ second? When $x=1.1$ seconds? When $x=1.2$ seconds?

(b) When is the height of the rock 15 meters? When is it 10 meters? When is it 5 meters?

(c) When does the rock strike the ground?

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Cost of Transatlantic Travel A Boeing 747 crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 miles per hour. The cost $C$ (in dollars) per passenger is given by

$$

C(x)=100+\frac{x}{10}+\frac{36,000}{x}

$$

where $x$ is the ground speed (airspeed $\pm$ wind $)$.

(a) What is the cost per passenger for quiescent (no wind) conditions?

(b) What is the cost per passenger with a head wind of 50 miles per hour?

(c) What is the cost per passenger with a tail wind of 100 miles per hour?

(d) What is the cost per passenger with a head wind of 100 miles per hour?

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Cross-sectional Area The cross-sectional area of a beam cut from a log with radius 1 foot is given by the function $A(x)=4 x \sqrt{1-x^{2}},$ where $x$ represents the length, in feet of half the base of the beam. See the figure. Determine the cross-sectional area of the beam if the length of half the base of the beam is as follows:

(a) One-third of a foot

(b) One-half of a foot

(c) Two-thirds of a foot

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Economics The participation rate is the number of people in the labor force divided by the civilian population (excludes military). Let $L(x)$ represent the size of the labor force in year $x,$ and $P(x)$ represent the civilian population in year $x$. Determine a function that represents the participation rate $R$ as a function of $x$.

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Crimes Suppose that $V(x)$ represents the number of violent crimes committed in year $x$ and $P(x)$ represents the number of property crimes committed in year $x$. Determine a function $T$ that represents the combined total of violent crimes and property crimes in year $x$.

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Health Care Suppose that $P(x)$ represents the percentage of income spent on health care in year $x$ and $I(x)$ represents income in year $x$. Find a function $H$ that represents total health care expenditures in year $x$.

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Income Tax Suppose that $I(x)$ represents the income of an individual in year $x$ before taxes and $T(x)$ represents the individual's tax bill in year $x .$ Find a function $N$ that represents the individual's net income (income after taxes) in year $x$.

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Profit Function Suppose that the revenue $R$, in dollars, from selling $x$ smartphones, in hundreds, is

$$

R(x)=-1.2 x^{2}+220 x

$$

The cost $C,$ in dollars, of selling $x$ smartphones, in hundreds, is $C(x)=0.05 x^{3}-2 x^{2}+65 x+500$

(a) Find the profit function, $P(x)=R(x)-C(x)$.

(b) Find the profit if $x=15$ hundred smartphones are sold.

(c) Interpret $P(15)$

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Population as a Function of Age The function

$$

P=P(a)=0.027 a^{2}-6.530 a+363.804

$$

represents the population $P$ (in millions) of Americans who are at least $a$ years old in 2015 .

(a) Identify the dependent and independent variables.

(b) Evaluate $P(20)$. Explain the meaning of $P(20)$.

(c) Evaluate $P(0)$. Explain the meaning of $P(0)$.

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Stopping Distance When the driver of a vehicle observes an impediment, the total stopping distance involves both the reaction distance $R$ (the distance the vehicle travels while the driver moves his or her foot to the brake pedal) and the braking distance $B$ (the distance the vehicle travels once the brakes are applied). For a car traveling at a speed of $v$ miles per hour, the reaction distance $R,$ in feet, can be estimated by $R(v)=2.2 v .$ Suppose that the braking distance $B,$ in feet, for a car is given by $B(v)=0.05 v^{2}+0.4 v-15$

(a) Find the stopping distance function

$$

D(v)=R(v)+B(v)

$$

(b) Find the stopping distance if the car is traveling at a speed of $60 \mathrm{mph}$

(c) Interpret $D(60)$

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Some functions $f$ have the property that

$$

f(a+b)=f(a)+f(b)

$$

for all real numbers $a$ and $b$. Which of the following functions have this property?

(a) $h(x)=2 x$

(b) $g(x)=x^{2}$

(c) $F(x)=5 x-2$

(d) $G(x)=\frac{1}{x}$

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Challenge Problem Find the difference quotient of the function $f(x)=\sqrt[3]{x}$ (Hint: Factor using $a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)$ with $a=\sqrt[3]{x+h}$ and $b=\sqrt[3]{x} .)$

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Challenge Problem If $f\left(\frac{x+4}{5 x-4}\right)=3 x^{2}-2,$ find $f(1)$

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Challenge Problem Find the domain of $f(x)=\sqrt{\frac{x^{2}+1}{7-|3 x-1|}}$

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Are the functions $f(x)=x-1$ and $g(x)=\frac{x^{2}-1}{x+1}$ the same? Explain.

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Find a function $H$ that multiplies a number $x$ by 3 and then subtracts the cube of $x$ and divides the result by your age.

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Investigate when, historically, the use of the function notation $y=f(x)$ first appeared.

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Problems 127-135 are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

List the intercepts and test for symmetry the graph of

$$

(x+12)^{2}+y^{2}=16

$$

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Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

Determine which of the given points are on the graph of the equation $y=3 x^{2}-8 \sqrt{x}$.

Points: (-1,-5),(4,32),(9,171)

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Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

How many pounds of lean hamburger that is $7 \%$ fat must be mixed with 12 pounds of ground chuck that is $20 \%$ fat to have a hamburger mixture that is $15 \%$ fat?

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Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

Solve $x^{3}-9 x=2 x^{2}-18$

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Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

Given $a+b x=a c+d,$ solve for $a$

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Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

Rotational Inertia The rotational inertia of an object varies directly with the square of the perpendicular distance from the object to the axis of rotation. If the rotational inertia is $0.4 \mathrm{~kg} \cdot \mathrm{m}^{2}$ when the perpendicular distance is $0.6 \mathrm{~m},$ what is the rotational inertia of the object if the perpendicular distance is $1.5 \mathrm{~m} ?$

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Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

Find the slope of a line perpendicular to the line

$$

3 x-10 y=12

$$

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Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.

Simplify $\frac{\left(4 x^{2}-7\right) \cdot 3-(3 x+5) \cdot 8 x}{\left(4 x^{2}-7\right)^{2}}$

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