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College Algebra 11th

Michael Sullivan

Chapter 3

Functions and Their Graphs

Educators


Problem 1

The inequality $-1<x<3$ can be written in interval notation as ________.

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Problem 2

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

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Problem 3

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

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Problem 4

Solve the inequality: $3-2 x>5 .$ Graph the solution set.

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Problem 5

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

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Problem 6

A quotient is considered rationalized if its denominator has no ______.

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Problem 7

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

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Problem 8

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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Problem 9

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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Problem 10

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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Problem 11

True or False Every relation is a function.

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Problem 12

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

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Problem 13

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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Problem 14

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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Problem 15

The expression $\frac{f(x+h)-f(x)}{h}$ is called the _______ of $f$.

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Problem 16

When written as $y=f(x),$ a function is said to be defined ______.

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Problem 17

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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Problem 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.
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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Problem 19

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

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Problem 20

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

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Problem 21

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

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Problem 22

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

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Problem 23

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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Problem 24

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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Problem 25

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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Problem 26

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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Problem 27

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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Problem 28

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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Problem 29

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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Problem 30

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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Problem 31

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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Problem 32

Determine whether the equation defines $y$ as a function of $x .$
$$
y=x^{3}
$$

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Problem 33

Determine whether the equation defines $y$ as a function of $x .$
$$
y=\frac{1}{x}
$$

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Problem 34

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

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Problem 35

Determine whether the equation defines $y$ as a function of $x .$
$$
x^{2}=8-y^{2}
$$

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Problem 36

Determine whether the equation defines $y$ as a function of $x .$
$$
y=\pm \sqrt{1-2 x}
$$

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Problem 37

Determine whether the equation defines $y$ as a function of $x .$
$$
x=y^{2}
$$

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Problem 38

Determine whether the equation defines $y$ as a function of $x .$
$$
x+y^{2}=1
$$

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Problem 39

Determine whether the equation defines $y$ as a function of $x .$
$$
y=\sqrt[3]{x}
$$

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Problem 40

Determine whether the equation defines $y$ as a function of $x .$
$$
y=\frac{3 x-1}{x+2}
$$

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Problem 41

Determine whether the equation defines $y$ as a function of $x .$
$$
|y|=2 x+3
$$

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Problem 41

Determine whether the equation defines $y$ as a function of $x .$
$$
|y|=2 x+3
$$

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Problem 42

Determine whether the equation defines $y$ as a function of $x .$
$$
x^{2}-4 y^{2}=1
$$

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Problem 43

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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Problem 44

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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Problem 45

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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Problem 46

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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Problem 47

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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Problem 48

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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Problem 49

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

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Problem 51

In Problems 51-70, find the domain of each function.
$$
f(x)=-5 x+4
$$

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Problem 52

Find the domain of each function.
$$
f(x)=x^{2}+2
$$

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Problem 53

Find the domain of each function.
$$
f(x)=\frac{x+1}{2 x^{2}+8}
$$

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Problem 54

Find the domain of each function.
$$
f(x)=\frac{x^{2}}{x^{2}+1}
$$

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Problem 55

Find the domain of each function.
$$
g(x)=\frac{x}{x^{2}-16}
$$

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Problem 56

Find the domain of each function.
$$
h(x)=\frac{2 x}{x^{2}-4}
$$

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Problem 57

Find the domain of each function.
$$
F(x)=\frac{x-2}{x^{3}+x}
$$

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Problem 58

Find the domain of each function.
$$
G(x)=\frac{x+4}{x^{3}-4 x}
$$

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Problem 59

Find the domain of each function.
$$
h(x)=\sqrt{3 x-12}
$$

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Problem 60

Find the domain of each function.
$$
G(x)=\sqrt{1-x}
$$

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Problem 61

Find the domain of each function.
$$
p(x)=\frac{x}{|2 x+3|-1}
$$

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Problem 62

Find the domain of each function.
$$
f(x)=\frac{x-1}{|3 x-1|-4}
$$

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Problem 63

Find the domain of each function.
$$
f(x)=\frac{x}{\sqrt{x-4}}
$$

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Problem 64

Find the domain of each function.
$$
f(x)=\frac{-x}{\sqrt{-x-2}}
$$

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Problem 65

Find the domain of each function.
$$
P(t)=\frac{\sqrt{t-4}}{3 t-21}
$$

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Problem 66

Find the domain of each function.
$$
h(z)=\frac{\sqrt{z+3}}{z-2}
$$

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Problem 67

Find the domain of each function.
$$
f(x)=\sqrt[3]{5 x-4}
$$

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Problem 68

Find the domain of each function.
$$
g(t)=-t^{2}+\sqrt[3]{t^{2}+7 t}
$$

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Problem 69

Find the domain of each function.
$$
M(t)=\sqrt[5]{\frac{t+1}{t^{2}-5 t-14}}
$$

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Problem 70

Find the domain of each function.
$$
N(p)=\sqrt[5]{\frac{p}{2 p^{2}-98}}
$$

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Problem 71

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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Problem 72

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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Problem 73

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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Problem 74

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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Problem 75

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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Problem 77

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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Problem 78

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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Problem 79

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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Problem 80

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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Problem 81

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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Problem 82

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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Problem 83

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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Problem 84

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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Problem 85

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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Problem 86

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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Problem 87

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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Problem 88

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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Problem 89

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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Problem 90

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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Problem 91

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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Problem 92

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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Problem 93

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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Problem 94

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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Problem 95

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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Problem 96

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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Problem 97

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

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Problem 99

If $f(x)=x^{2}-2 x+3,$ find the value(s) of $x$ so that $f(x)=11$

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Problem 100

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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Problem 101

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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Problem 102

If $f(x)=3 x^{2}-B x+4$ and $f(-1)=12,$ what is the value of $B ?$

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Problem 103

If $f(x)=\frac{3 x+8}{2 x-A}$ and $f(0)=2,$ what is the value of $A ?$

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Problem 104

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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Problem 105

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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Problem 106

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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Problem 107

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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Problem 108

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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Problem 109

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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Problem 110

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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Problem 111

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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Problem 112

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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Problem 113

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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Problem 114

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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Problem 115

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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Problem 116

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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Problem 117

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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Problem 118

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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Problem 119

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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Problem 120

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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Problem 121

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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Problem 122

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

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Problem 123

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

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Problem 124

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

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Problem 125

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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Problem 126

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

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Problem 127

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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Problem 128

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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Problem 129

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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Problem 130

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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Problem 131

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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Problem 132

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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Problem 133

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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Problem 134

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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Problem 135

Determine the degree of the polynomial
$$
9 x^{2}(3 x-5)(5 x+1)^{4}
$$

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