Intermediate Algebra for College Students

Robert Blitzer

Chapter 6

Rational Expressions, Functions, and Equations 403 - all with Video Answers

Educators

Section 1

Rational Expressions and Functions: Multiplying and Dividing

Problem 1

Use the given rational function to find the indicated function values. If a function value does not exist, so state.
$$
f(x)=\frac{x^{2}-9}{x+3} ; f(-2), f(0), f(5)
$$

AG

Ankit G.

Numerade Educator

Problem 2

Use the given rational function to find the indicated function values. If a function value does not exist, so state.
$$
f(x)=\frac{x^{2}-16}{x+4} ; f(-2), f(0), f(5)
$$

AG

Ankit G.

Numerade Educator

Problem 3

Use the given rational function to find the indicated function values. If a function value does not exist, so state.
$$
f(x)=\frac{x^{2}-2 x-3}{4-x} ; f(-1), f(4), f(6)
$$

AG

Ankit G.

Numerade Educator

Problem 4

Use the given rational function to find the indicated function values. If a function value does not exist, so state.
$$
f(x)=\frac{x^{2}-3 x-4}{3-x} ; f(-1), f(3), f(5)
$$

AG

Ankit G.

Numerade Educator

Problem 5

Use the given rational function to find the indicated function values. If a function value does not exist, so state.
$$
f(x)=\frac{x^{2}-3 x-4}{3-x} ; f(-1), f(3), f(5)
$$

AG

Ankit G.

Numerade Educator

Problem 6

Use the given rational function to find the indicated function values. If a function value does not exist, so state.
$$
g(t)=\frac{2 t^{3}-1}{t^{2}+4} ; g(-1), g(0), g(2)
$$

AG

Ankit G.

Numerade Educator

Problem 7

Find the domain of the given rational function.
$$
f(x)=\frac{x-2}{x-5}
$$

AG

Ankit G.

Numerade Educator

Problem 8

Find the domain of the given rational function.
$$
f(x)=\frac{x-3}{x-6}
$$

AG

Ankit G.

Numerade Educator

Problem 9

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

AG

Ankit G.

Numerade Educator

Problem 10

Find the domain of the given rational function.
$$
f(x)=\frac{x-5}{(x-2)(x+4)}
$$

AG

Ankit G.

Numerade Educator

Problem 11

Find the domain of the given rational function.
$$
f(x)=\frac{2 x}{(x+5)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 12

Find the domain of the given rational function.
$$
f(x)=\frac{2 x}{(x+7)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 13

Find the domain of the given rational function.
$$
f(x)=\frac{3 x}{x^{2}-8 x+15}
$$

AG

Ankit G.

Numerade Educator

Problem 14

Find the domain of the given rational function.
$$
f(x)=\frac{3 x}{x^{2}-13 x+36}
$$

AG

Ankit G.

Numerade Educator

Problem 15

Find the domain of the given rational function.
$$
f(x)=\frac{(x-1)^{2}}{3 x^{2}-2 x-8}
$$

AG

Ankit G.

Numerade Educator

Problem 16

Find the domain of the given rational function.
$$
f(x)=\frac{(x-1)^{2}}{4 x^{2}-13 x+3}
$$

AG

Ankit G.

Numerade Educator

Problem 17

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
Find $f(4)$

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Problem 18

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
Find $f(1)$

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Problem 19

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
What is the domain of $f ?$ What is the range of $f ?$

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Problem 20

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
What are the equations of the vertical asymptotes of the graph of $f ?$

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Problem 21

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
Describe the end behavior of the graph at the far left. What is the equation of the horizontal asymptote?

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Problem 22

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
Describe the end behavior of the graph at the far right. What is the equation of the horizontal asymptote?

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Problem 23

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
Explain how the graph shows that $f(-2)$ does not exist.

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Problem 24

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
Explain how the graph shows that $f(2)$ does not exist.

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Problem 25

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
How can you tell that this is not the graph of a polynomial function?

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Problem 26

The graph of a rational function, $f,$ is shown in the figure. Use the graph to solve Exercises $17-26 .$
List two real numbers that are not function values of $f$.

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Problem 27

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{2}-4}{x-2}
$$

Amy J.

Numerade Educator

Problem 28

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{2}-25}{x-5}
$$

AG

Ankit G.

Numerade Educator

Problem 29

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x+2}{x^{2}-x-6}
$$

Amy J.

Numerade Educator

Problem 30

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x+1}{x^{2}-2 x-3}
$$

Amy J.

Numerade Educator

Problem 31

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{4 x+20}{x^{2}+5 x}
$$

AG

Ankit G.

Numerade Educator

Problem 32

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{5 x+30}{x^{2}+6 x}
$$

AG

Ankit G.

Numerade Educator

Problem 33

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{4 y-20}{y^{2}-25}
$$

AG

Ankit G.

Numerade Educator

Problem 34

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{6 y-42}{y^{2}-49}
$$

AG

Ankit G.

Numerade Educator

Problem 35

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{3 x-5}{25-9 x^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 36

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{5 x-2}{4-25 x^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 37

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{y^{2}-49}{y^{2}-14 y+49}
$$

AG

Ankit G.

Numerade Educator

Problem 38

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{y^{2}-9}{y^{2}-6 y+9}
$$

AG

Ankit G.

Numerade Educator

Problem 39

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{2}+7 x-18}{x^{2}-3 x+2}
$$

AG

Ankit G.

Numerade Educator

Problem 40

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{2}-4 x-5}{x^{2}+5 x+4}
$$

AG

Ankit G.

Numerade Educator

Problem 41

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{3 x+7}{3 x+10}
$$

Amy J.

Numerade Educator

Problem 42

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{2 x+3}{2 x+5}
$$

Amy J.

Numerade Educator

Problem 43

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{2}-x-12}{16-x^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 44

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{2}-7 x+12}{9-x^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 45

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{2}+3 x y-10 y^{2}}{3 x^{2}-7 x y+2 y^{2}}
$$

Amy J.

Numerade Educator

Problem 46

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{2}+2 x y-3 y^{2}}{2 x^{2}+5 x y-3 y^{2}}
$$

Amy J.

Numerade Educator

Problem 47

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{3}-8}{x^{2}-4}
$$

AG

Ankit G.

Numerade Educator

Problem 48

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{3}-1}{x^{2}-1}
$$

AG

Ankit G.

Numerade Educator

Problem 49

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{3}+4 x^{2}-3 x-12}{x+4}
$$

Amy J.

Numerade Educator

Problem 50

Simplify each rational expression. If the rational expression cannot be simplified, so state.
$$
\frac{x^{3}-2 x^{2}+x-2}{x-2}
$$

Amy J.

Numerade Educator

Problem 51

Multiply as indicated.
$$
\frac{x-3}{x+7} \cdot \frac{3 x+21}{2 x-6}
$$

AG

Ankit G.

Numerade Educator

Problem 52

Multiply as indicated.
$$
\frac{x-2}{x+3} \cdot \frac{2 x+6}{5 x-10}
$$

AG

Ankit G.

Numerade Educator

Problem 53

Multiply as indicated.
$$
\frac{x^{2}-49}{x^{2}-4 x-21} \cdot \frac{x+3}{x}
$$

Amy J.

Numerade Educator

Problem 54

Multiply as indicated.
$$
\frac{x^{2}-25}{x^{2}-3 x-10} \cdot \frac{x+2}{x}
$$

Amy J.

Numerade Educator

Problem 55

Multiply as indicated.
$$
\frac{x^{2}-9}{x^{2}-x-6} \cdot \frac{x^{2}+5 x+6}{x^{2}+x-6}
$$

AG

Ankit G.

Numerade Educator

Problem 56

Multiply as indicated.
$$
\frac{x^{2}-1}{x^{2}-4} \cdot \frac{x^{2}-5 x+6}{x^{2}-2 x-3}
$$

AG

Ankit G.

Numerade Educator

Problem 57

Multiply as indicated.
$$
\frac{x^{2}+4 x+4}{x^{2}+8 x+16} \cdot \frac{(x+4)^{3}}{(x+2)^{3}}
$$

AG

Ankit G.

Numerade Educator

Problem 58

Multiply as indicated.
$$
\frac{x^{2}-2 x+1}{x^{2}-4 x+4} \cdot \frac{(x-2)^{3}}{(x-1)^{3}}
$$

AG

Ankit G.

Numerade Educator

Problem 59

Multiply as indicated.
$$
\frac{8 y+2}{y^{2}-9} \cdot \frac{3-y}{4 y^{2}+y}
$$

AG

Ankit G.

Numerade Educator

Problem 60

Multiply as indicated.
$$
\frac{6 y+2}{y^{2}-1} \cdot \frac{1-y}{3 y^{2}+y}
$$

AG

Ankit G.

Numerade Educator

Problem 61

Multiply as indicated.
$$
\frac{y^{3}-8}{y^{2}-4} \cdot \frac{y+2}{2 y}
$$

AG

Ankit G.

Numerade Educator

Problem 62

Multiply as indicated.
$$
\frac{y^{2}+6 y+9}{y^{3}+27} \cdot \frac{1}{y+3}
$$

AG

Ankit G.

Numerade Educator

Problem 63

Multiply as indicated.
$$
(x-3) \cdot \frac{x^{2}+x+1}{x^{2}-5 x+6}
$$

AG

Ankit G.

Numerade Educator

Problem 64

Multiply as indicated.
$$
(x+1) \cdot \frac{x+2}{x^{2}+7 x+6}
$$

AG

Ankit G.

Numerade Educator

Problem 65

Multiply as indicated.
$$
\frac{x^{2}+x y}{x^{2}-y^{2}} \cdot \frac{4 x-4 y}{x}
$$

AG

Ankit G.

Numerade Educator

Problem 66

Multiply as indicated.
$$
\frac{x^{2}-y^{2}}{x} \cdot \frac{x^{2}+x y}{x+y}
$$

Amy J.

Numerade Educator

Problem 67

Multiply as indicated.
$$
\frac{x^{2}+2 x y+y^{2}}{x^{2}-2 x y+y^{2}} \cdot \frac{4 x-4 y}{3 x+3 y}
$$

Amy J.

Numerade Educator

Problem 68

Multiply as indicated.
$$
\frac{2 x^{2}-3 x y-2 y^{2}}{3 x^{2}-4 x y+y^{2}} \cdot \frac{3 x^{2}-2 x y-y^{2}}{x^{2}+x y-6 y^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 69

Multiply as indicated.
$$
\frac{4 a^{2}+2 a b+b^{2}}{2 a+b} \cdot \frac{4 a^{2}-b^{2}}{8 a^{3}-b^{3}}
$$

AG

Ankit G.

Numerade Educator

Problem 70

Multiply as indicated.
$$
\frac{27 a^{3}-8 b^{3}}{b^{2}-b-6} \cdot \frac{b c-b-3 c+3}{3 a c-2 b c-3 a+2 b}
$$

AG

Ankit G.

Numerade Educator

Problem 71

Multiply as indicated.
$$
\frac{10 z^{2}+13 z-3}{3 z^{2}-8 z+5} \cdot \frac{2 z^{2}-3 z-2 z+3}{25 z^{2}-10 z+1} \cdot \frac{15 z^{2}-28 z+5}{4 z^{2}-9}
$$

AG

Ankit G.

Numerade Educator

Problem 72

Multiply as indicated.
$$
\frac{2 z^{2}-2 z-12}{z^{2}-49} \cdot \frac{4 z^{2}-1}{2 z^{2}+5 z+2} \cdot \frac{2 z^{2}-13 z-7}{2 z^{2}-7 z+3}
$$

AG

Ankit G.

Numerade Educator

Problem 73

Divide as indicated.
$$
\frac{x+5}{7} \div \frac{4 x+20}{9}
$$

Amy J.

Numerade Educator

Problem 74

Divide as indicated.
$$
\frac{x+1}{3} \div \frac{3 x+3}{7}
$$

Amy J.

Numerade Educator

Problem 75

Divide as indicated.
$$
\frac{4}{y-6} \div \frac{40}{7 y-42}
$$

AG

Ankit G.

Numerade Educator

Problem 76

Divide as indicated.
$$
\frac{7}{y-5} \div \frac{28}{3 y-15}
$$

AG

Ankit G.

Numerade Educator

Problem 77

Divide as indicated.
$$
\frac{x^{2}-2 x}{15} \div \frac{x-2}{5}
$$

AG

Ankit G.

Numerade Educator

Problem 78

Divide as indicated.
$$
\frac{x^{2}-x}{15} \div \frac{x-1}{5}
$$

AG

Ankit G.

Numerade Educator

Problem 79

Divide as indicated.
$$
\frac{y^{2}-25}{2 y-2} \div \frac{y^{2}+10 y+25}{y^{2}+4 y-5}
$$

AG

Ankit G.

Numerade Educator

Problem 80

Divide as indicated.
$$
\frac{y^{2}+y}{y^{2}-4} \div \frac{y^{2}-1}{y^{2}+5 y+6}
$$

AG

Ankit G.

Numerade Educator

Problem 81

Divide as indicated.
$$
\left(x^{2}-16\right) \div \frac{x^{2}+3 x-4}{x^{2}+4}
$$

AG

Ankit G.

Numerade Educator

Problem 82

Divide as indicated.
$$
\left(x^{2}+4 x-5\right) \div \frac{x^{2}-25}{x+7}
$$

AG

Ankit G.

Numerade Educator

Problem 83

Divide as indicated.
$$
\frac{y^{2}-4 y-21}{y^{2}-10 y+25} \div \frac{y^{2}+2 y-3}{y^{2}-6 y+5}
$$

AG

Ankit G.

Numerade Educator

Problem 84

Divide as indicated.
$$
\frac{y^{2}+4 y-21}{y^{2}+3 y-28} \div \frac{y^{2}+14 y+48}{y^{2}+4 y-32}
$$

Amy J.

Numerade Educator

Problem 85

Divide as indicated.
$$
\frac{8 x^{3}-1}{4 x^{2}+2 x+1} \div \frac{x-1}{(x-1)^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 86

Divide as indicated.
$$
\frac{x^{2}-9}{x^{3}-27} \div \frac{x^{2}+6 x+9}{x^{2}+3 x+9}
$$

AG

Ankit G.

Numerade Educator

Problem 87

Divide as indicated.
$$
\frac{x^{2}-4 y^{2}}{x^{2}+3 x y+2 y^{2}} \div \frac{x^{2}-4 x y+4 y^{2}}{x+y}
$$

Amy J.

Numerade Educator

Problem 88

Divide as indicated.
$$
\frac{x y-y^{2}}{x^{2}+2 x+1} \div \frac{2 x^{2}+x y-3 y^{2}}{2 x^{2}+5 x y+3 y^{2}}
$$

Amy J.

Numerade Educator

Problem 89

Divide as indicated.
$$
\frac{x^{4}-y^{8}}{x^{2}+y^{4}} \div \frac{x^{2}-y^{4}}{3 x^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 90

Divide as indicated.
$$
\frac{x^{4}-y^{8}}{x^{2}+y^{4}} \div \frac{x^{2}-y^{4}}{3 x^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 91

Perform the indicated operation or operations.
$$
\frac{x^{3}-4 x^{2}+x-4}{2 x^{3}-8 x^{2}+x-4} \cdot \frac{2 x^{3}+2 x^{2}+x+1}{x^{4}-x^{3}+x^{2}-x}
$$

AG

Ankit G.

Numerade Educator

Problem 92

Perform the indicated operation or operations.
$$
\frac{y^{3}+y^{2}+y z^{2}+z^{2}}{y^{3}+y+y^{2}+1} \cdot \frac{y^{3}+y+y^{2} z+z}{2 y^{2}+2 y z-y z^{2}-z^{3}}
$$

AG

Ankit G.

Numerade Educator

Problem 93

Perform the indicated operation or operations.
$$
\frac{a x-a y+3 x-3 y}{x^{3}+y^{3}} \div \frac{a b+3 b+a c+3 c}{x y-x^{2}-y^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 93

Perform the indicated operation or operations.
$$
\frac{a x-a y+3 x-3 y}{x^{3}+y^{3}} \div \frac{a b+3 b+a c+3 c}{x y-x^{2}-y^{2}}
$$

AG

Ankit G.

Numerade Educator

Problem 94

Perform the indicated operation or operations.
$$
\frac{a^{3}+b^{3}}{a c-a d-b c+b d} \div \frac{a b-a^{2}-b^{2}}{a c-a d+b c-b d}
$$

AG

Ankit G.

Numerade Educator

Problem 95

Perform the indicated operation or operations.
$$
\frac{a^{2} b+b}{3 a^{2}-4 a-20} \cdot \frac{a^{2}+5 a}{2 a^{2}+11 a+5} \div \frac{a b^{2}}{6 a^{2}-17 a-10}
$$

AG

Ankit G.

Numerade Educator

Problem 96

Perform the indicated operation or operations.
$$
\frac{a^{2}-8 a+15}{2 a^{3}-10 a^{2}} \cdot \frac{2 a^{2}+3 a}{3 a^{3}-27 a} \div \frac{14 a+21}{a^{2}-6 a-27}
$$

AG

Ankit G.

Numerade Educator

Problem 97

Perform the indicated operation or operations.
$$
\frac{a-b}{4 c} \div\left(\frac{b-a}{c} \div \frac{a-b}{c^{2}}\right)
$$

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Problem 98

Perform the indicated operation or operations.
$$
\left(\frac{a-b}{4 c} \div \frac{b-a}{c}\right) \div \frac{a-b}{c^{2}}
$$

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Problem 99

Find $\frac{f(a+h)-f(a)}{h}$ and simplify.
$$
f(x)=7 x-4
$$

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Ankit G.

Numerade Educator

Problem 100

Find $\frac{f(a+h)-f(a)}{h}$ and simplify.
$$
f(x)=-3 x+5
$$

AG

Ankit G.

Numerade Educator

Problem 101

Find $\frac{f(a+h)-f(a)}{h}$ and simplify.
$$
f(x)=x^{2}-5 x+3
$$

AG

Ankit G.

Numerade Educator

Problem 102

Find $\frac{f(a+h)-f(a)}{h}$ and simplify.
$$
f(x)=3 x^{2}-4 x+7
$$

AG

Ankit G.

Numerade Educator

Problem 103

Let
$$
f(x)=\frac{(x+2)^{2}}{1-2 x} \text { and } g(x)=\frac{x+2}{2 x-1} \text { . }
$$
$$
\text { Find }\left(\frac{f}{g}\right)(x) \text { and the domain of } \frac{f}{g} \text { . }
$$

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Problem 104

Let
$$
f(x)=\frac{(x+2)^{2}}{1-2 x} \text { and } g(x)=\frac{x+2}{2 x-1}
$$
$$
\text { Find }\left(\frac{g}{f}\right)(x) \text { and the domain of } \frac{g}{f}
$$

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Problem 105

The rational function
$$
f(x)=\frac{130 x}{100-x}
$$
models the cost, $f(x),$ in millions of dollars, to inoculate $x \%$ of the population against a particular strain of flu. The graph of the rational function is shown. Use the function's equation to solve Exercises $105-108 .$
Find and interpret $f(60)$. Identify your solution as a point on the graph.

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Problem 106

The rational function
$$
f(x)=\frac{130 x}{100-x}
$$
models the cost, $f(x),$ in millions of dollars, to inoculate $x \%$ of the population against a particular strain of flu. The graph of the rational function is shown. Use the function's equation to solve Exercises $105-108 .$
Find and interpret $f(80)$. Identify your solution as a point on the graph.

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Problem 107

The rational function
$$
f(x)=\frac{130 x}{100-x}
$$
models the cost, $f(x),$ in millions of dollars, to inoculate $x \%$ of the population against a particular strain of flu. The graph of the rational function is shown. Use the function's equation to solve Exercises $105-108 .$
What value of $x$ must be excluded from the rational function's domain? Does the cost model indicate that we can inoculate all of the population against the flu? Explain.

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Problem 108

The rational function
$$
f(x)=\frac{130 x}{100-x}
$$
models the cost, $f(x),$ in millions of dollars, to inoculate $x \%$ of the population against a particular strain of flu. The graph of the rational function is shown. Use the function's equation to solve Exercises $105-108 .$
What happens to the cost as $x$ approaches $100 \%$ ? How is this shown by the graph? Explain what this means.

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Problem 109

The function
$$
f(x)=\frac{6.5 x^{2}-20.4 x+234}{x^{2}+36}
$$
models the pH level, $f(x),$ of the human mouth $x$ minutes after a person eats food containing sugar. The graph of this function is shown in the figure. Use the function's equation or graph, as specified, to solve Exercises $109-112 .$
After eating sugar, when is the pH level the lowest? Use the function's equation to determine the pH level, to the nearest tenth, at this time.

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Problem 110

The function
$$
f(x)=\frac{6.5 x^{2}-20.4 x+234}{x^{2}+36}
$$
models the pH level, $f(x),$ of the human mouth $x$ minutes after a person eats food containing sugar. The graph of this function is shown in the figure. Use the function's equation or graph, as specified, to solve Exercises $109-112 .$
Use the graph to obtain a reasonable estimate, to the nearest tenth, of the pH level of the human mouth 42 minutes after a person eats food containing sugar.

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Problem 111

The function
$$
f(x)=\frac{6.5 x^{2}-20.4 x+234}{x^{2}+36}
$$
models the pH level, $f(x),$ of the human mouth $x$ minutes after a person eats food containing sugar. The graph of this function is shown in the figure. Use the function's equation or graph, as specified, to solve Exercises $109-112 .$
According to the graph, what is the normal pH level of the human mouth? What does the end behavior at the far right of the graph indicate in terms of the mouth's pH level over time?

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Problem 112

The function
$$
f(x)=\frac{6.5 x^{2}-20.4 x+234}{x^{2}+36}
$$
models the pH level, $f(x),$ of the human mouth $x$ minutes after a person eats food containing sugar. The graph of this function is shown in the figure. Use the function's equation or graph, as specified, to solve Exercises $109-112 .$
Use the graph to describe what happens to the $\mathrm{pH}$ level during the first hour.

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Problem 113

The rational function
$$
P(x)=\frac{100(x-1)}{x}
$$
models the percentage of smoking-related deaths among all deaths from a disease, $P(x),$ in terms of the disease's incidence ratio, $x .$ The graph of the rational function is shown. Use this function to solve Exercises $113-116 .$
Find $P(10) .$ Describe what this means in terms of the incidence ratio, $10,$ given in the table. Identify your solution as a point on the graph.

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Problem 114

The rational function
$$
P(x)=\frac{100(x-1)}{x}
$$
models the percentage of smoking-related deaths among all deaths from a disease, $P(x),$ in terms of the disease's incidence ratio, $x .$ The graph of the rational function is shown. Use this function to solve Exercises $113-116 .$
Find $P(9)$. Round to the nearest percent. Describe what this means in terms of the incidence ratio, $9,$ given in the table. Identify your solution as a point on the graph.

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Problem 115

The rational function
$$
P(x)=\frac{100(x-1)}{x}
$$
models the percentage of smoking-related deaths among all deaths from a disease, $P(x),$ in terms of the disease's incidence ratio, $x .$ The graph of the rational function is shown. Use this function to solve Exercises $113-116 .$
What is the horizontal asymptote of the graph? Describe what this means about the percentage of deaths caused by smoking with increasing incidence ratios.

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Problem 116

The rational function
$$
P(x)=\frac{100(x-1)}{x}
$$
models the percentage of smoking-related deaths among all deaths from a disease, $P(x),$ in terms of the disease's incidence ratio, $x .$ The graph of the rational function is shown. Use this function to solve Exercises $113-116 .$
According to the model and its graph, is there a disease for which all deaths are caused by smoking? Explain your answer.

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Problem 117

What is a rational expression? Give an example with your explanation.

Amy J.

Numerade Educator

Problem 118

What is a rational function? Provide an example.

AG

Ankit G.

Numerade Educator

Problem 119

What is the domain of a rational function?

Mike G.

Numerade Educator

Problem 120

If you are given the equation of a rational function, explain how to determine its domain.

AG

Ankit G.

Numerade Educator

Problem 121

Describe two ways the graph of a rational function differs from the graph of a polynomial function.

AG

Ankit G.

Numerade Educator

Problem 122

What is a vertical asymptote?

Michael T.

Numerade Educator

Problem 123

What is a horizontal asymptote?

Linh V.

Numerade Educator

Problem 124

Explain how to simplify a rational expression.

Amy J.

Numerade Educator

Problem 125

Explain how to simplify a rational expression with opposite factors in the numerator and the denominator

Amy J.

Numerade Educator

Problem 126

Explain how to simplify a rational expression with opposite factors in the numerator and the denominator.

Amy J.

Numerade Educator

Problem 127

Explain how to divide rational expressions.

Joanna Q.

Numerade Educator

Problem 128

Although your friend has a family history of heart disease, he smokes, on average, 25 cigarettes per day. He sees the table showing incidence ratios for heart disease (see Exercises $113-116$ ) and feels comfortable that they are less than $2,$ compared to 9 and 10 for lung cancer. He claims that all family deaths have been from heart disease and decides not to give up smoking. Use the given function and its graph to describe some additional information not given in the table that might influence his decision.

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Problem 129

Determine if the multiplication or division has been performed correctly by graphing the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side and then verify your correction using the graphing utility.
$$
\frac{x^{2}+x}{3 x} \cdot \frac{6 x}{x+1}=2 x
$$

AG

Ankit G.

Numerade Educator

Problem 130

Determine if the multiplication or division has been performed correctly by graphing the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side and then verify your correction using the graphing utility.
$$
\frac{x^{3}-25 x}{x^{2}-3 x-10} \cdot \frac{x+2}{x}=x+5
$$

AG

Ankit G.

Numerade Educator

Problem 131

Determine if the multiplication or division has been performed correctly by graphing the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side and then verify your correction using the graphing utility.
$$
\frac{x^{2}-9}{x+4} \div \frac{x-3}{x+4}=x-3
$$

AG

Ankit G.

Numerade Educator

Problem 132

Determine if the multiplication or division has been performed correctly by graphing the function on each side of the equation in the same viewing rectangle. If the graphs do not coincide, correct the expression on the right side and then verify your correction using the graphing utility.
$$
(x-5) \div \frac{2 x^{2}-11 x+5}{4 x^{2}-1}=2 x-1
$$

AG

Ankit G.

Numerade Educator

Problem 133

Use the table the domains that you determined for any two functions in Exercises $7-16$.

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Problem 134

a. Graph $f(x)=\frac{x^{2}-x-2}{x-2}$ and $g(x)=x+1$ in the same viewing rectangle. What do you observe?
b. Simplify the formula in the definition of function $f .$ Do $f$ and $g$ represent exactly the same function? Explain.
c. Display the graphs of $f$ and $g$ separately in the utility's viewing rectangle trace along each of the curves until you get to $x=2 .$ What difference do you observe? What does this mean?

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Problem 135

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
I cannot simplify, multiply, or divide rational expressions without knowing how to factor polynomials.

AG

Ankit G.

Numerade Educator

Problem 136

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
I simplified $\frac{2(x+2)-5(x+1)}{(x+2)(x+1)}$ by dividing the numerator and the denominator by $(x+2)(x+1)$.

AG

Ankit G.

Numerade Educator

Problem 137

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
The values to exclude from the domain of $f(x)=\frac{x-3}{x-7}$ are 3 and 7.

AG

Ankit G.

Numerade Educator

Problem 138

Determine whether each statement makes sense or does not make sense, and explain your reasoning.
When performing the division
$$
\frac{3 x}{x+2} \div \frac{(x+2)^{2}}{x-4}
$$
I began by dividing the numerator and the denominator by the common factor, $x+2$.

AG

Ankit G.

Numerade Educator

Problem 139

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$
\frac{x^{2}-25}{x-5}=x-5
$$

Anna M.

Numerade Educator

Problem 140

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$
\frac{x^{2}+7}{7}=x^{2}+1
$$

AG

Ankit G.

Numerade Educator

Problem 141

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
$$
\begin{aligned}
&\text { The domain of } f(x)=\frac{7}{x(x-3)+5(x-3)} \text { is }\\
&(-\infty, 3) \cup(3, \infty)
\end{aligned}
$$

AG

Ankit G.

Numerade Educator

Problem 142

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The restrictions on the values of $x$ when performing the division
$$\frac{f(x)}{g(x)} \div \frac{h(x)}{k(x)}$$
are $g(x) \neq 0, k(x) \neq 0,$ and $h(x) \neq 0 .$

AG

Ankit G.

Numerade Educator

Problem 143

Graph: $f(x)=\frac{x^{2}-x-2}{x-2}$.

AG

Ankit G.

Numerade Educator

Problem 144

Simplify: $\frac{6 x^{3 n}+6 x^{2 n} y^{n}}{x^{2 n}-y^{2 n}}$

AG

Ankit G.

Numerade Educator

Problem 145

Divide: $\frac{y^{2 n}-1}{y^{2 n}+3 y^{n}+2} \div \frac{y^{2 n}+y^{n}-12}{y^{2 n}-y^{n}-6}$

AG

Ankit G.

Numerade Educator

Problem 146

Solve for $x$ in terms of $a$ and write the resulting rational expression in simplified form. Include any necessary restrictions on $a$.
$$
a^{2}(x-1)=4(x-1)+5 a+10
$$

AG

Ankit G.

Numerade Educator

Problem 147

Graph: $4 x-5 y \geq 20$.

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Problem 148

Multiply: $(2 x-5)\left(x^{2}-3 x-6\right)$.

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Problem 149

Simplify: $\left(\frac{a b^{-3} c^{-4}}{4 a^{5} b^{10} c^{-3}}\right)^{-2}$

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Problem 150

Perform the indicated operation. Where possible, reduce the answer to its lowest terms.
$$
\frac{7}{10}-\frac{3}{10}
$$

AG

Ankit G.

Numerade Educator

Problem 151

Perform the indicated operation. Where possible, reduce the answer to its lowest terms.
$$
\frac{1}{2}+\frac{2}{3}
$$

James K.

Numerade Educator

Problem 152

Perform the indicated operation. Where possible, reduce the answer to its lowest terms.
$$
\frac{7}{15}-\frac{3}{10}
$$

AG

Ankit G.

Numerade Educator