Elementary and Intermediate Algebra

Marvin L. Bittinger, David J. Ellenbogen,Barbara L. Johnson

Chapter 7

Rational Expressions, Equations, and Functions

Educators


Problem 1

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$
f(x)=\frac{2-x}{x-5}
$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

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Problem 2

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$
f(x)=\frac{x+5}{x+2}
$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

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Problem 3

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$
g(x)=\frac{x-3}{(x-2)(x-5)}
$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

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Problem 4

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$
g(x)=\frac{x+3}{(x+2)(x-5)}
$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

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Problem 5

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$
-h(x)=\frac{(x-2)(x-3)}{x+3}
$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

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Problem 6

In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.
$$
f(x)=\frac{(x+2)(x+3)}{x-3}
$$
a) $\{x | x \neq-5, x \neq 2\}$
b) $\{x | x \neq 3\}$
c) $\{x | x \neq-2\}$
d) $\{x | x \neq-3\}$
e) $\{x | x \neq 5\}$
f) $\{x | x \neq-2, x \neq 5\}$
g) $\{x | x \neq 2\}$
h) $\{x | x \neq-2, x \neq-5\}$
i) $\quad\{x | x \neq 2, x \neq 5\}$
j) $\{x | x \neq-5\}$

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Problem 7

Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by
$$
H(t)=\frac{t^{2}+3 t}{2 t+3}
$$
can be used to determine how long it would take if they worked together.
How long will it take them, working together, to complete a day's orders if Molly can process the orders alone in 5 hr?

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Problem 8

Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by
$$
H(t)=\frac{t^{2}+3 t}{2 t+3}
$$
can be used to determine how long it would take if they worked together.
How long will it take them, working together, to complete a day's orders if Molly can process the srders alone in 7 hr?

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Problem 9

Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by
$$
H(t)=\frac{t^{2}+3 t}{2 t+3}
$$
can be used to determine how long it would take if they worked together.

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Problem 10

For each rational function, find the function values indicated, provided the value exists.
$$
v(t)=\frac{4 t^{2}-5 t+2}{t+3} ; v(0), v(-2), v(7)
$$

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Problem 11

For each rational function, find the function values indicated, provided the value exists.
$$
g(x)=\frac{2 x^{3}-9}{x^{2}-4 x+4} ; g(0), g(2), g(-1)
$$

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Problem 12

For each rational function, find the function values indicated, provided the value exists.
$$
r(t)=\frac{t^{2}-5 t+4}{t^{2}-9} ; r(1), r(2), r(-3)
$$

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Problem 13

List all numbers for which each rational expression is undefined.
$$
\frac{25}{-7 x}
$$

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Problem 14

List all numbers for which each rational expression is undefined.
$$
\frac{14}{-5 y}
$$

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Problem 15

List all numbers for which each rational expression is undefined.
$$
\frac{t-3}{t+8}
$$

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Problem 16

List all numbers for which each rational expression is undefined.
$$
\frac{a-8}{a+7}
$$

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Problem 17

List all numbers for which each rational expression is undefined.
$$
\frac{a}{3 a-12}
$$

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Problem 18

List all numbers for which each rational expression is undefined.
$$
\frac{x^{2}}{4 x-12}
$$

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Problem 19

List all numbers for which each rational expression is undefined.
$$
\frac{x^{2}-16}{x^{2}-3 x-28}
$$

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Problem 20

List all numbers for which each rational expression is undefined.
$$
\frac{p^{2}-9}{p^{2}-7 p+10}
$$

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Problem 21

List all numbers for which each rational expression is undefined.
$$
\frac{m^{3}-2 m}{m^{2}-25}
$$

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Problem 22

List all numbers for which each rational expression is undefined.
$$
\frac{7-3 x+x^{2}}{49-x^{2}}
$$

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Problem 23

Simplify by removing a factor equal to 1.
$$
\frac{15 x}{5 x^{2}}
$$

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Problem 24

Simplify by removing a factor equal to 1.
$$
\frac{7 a^{3}}{21 a}
$$

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Problem 25

Simplify by removing a factor equal to 1.
$$
\frac{18 t^{3} w^{2}}{27 t^{7} w}
$$

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Problem 26

Simplify by removing a factor equal to 1.
$$
\frac{8 y^{5} z}{4 y^{9} z^{3}}
$$

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Problem 27

Simplify by removing a factor equal to 1.
$$
\frac{2 a-10}{2}
$$

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Problem 28

Simplify by removing a factor equal to 1.
$$
\frac{3 a+12}{3}
$$

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Problem 29

Simplify by removing a factor equal to 1.
$$
\frac{3 x^{2}-12 x}{3 x^{2}+15 x}
$$

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Problem 30

Simplify by removing a factor equal to 1.
$$
\frac{4 y^{2}-20 y}{4 y^{2}+12 y}
$$

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Problem 31

Simplify by removing a factor equal to 1.
$$
\frac{6 a^{2}-3 a}{7 a^{2}-7 a}
$$

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Problem 32

Simplify by removing a factor equal to 1.
$$
\frac{3 m^{2}+3 m}{6 m^{2}+9 m}
$$

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Problem 33

Simplify by removing a factor equal to 1.
$$
\frac{3 a-1}{2-6 a}
$$

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Problem 34

Simplify by removing a factor equal to 1.
$$
\frac{6-5 a}{10 a-12}
$$

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Problem 35

Simplify by removing a factor equal to 1.
$$
\frac{3 a^{2}+9 a-12}{6 a^{2}-30 a+24}
$$

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Problem 36

Simplify by removing a factor equal to 1.
$$
\frac{2 t^{2}-6 t+4}{4 t^{2}+12 t-16}
$$

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Problem 37

Simplify by removing a factor equal to 1.
$$
\frac{x^{2}+8 x+16}{x^{2}-16}
$$

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Problem 38

Simplify by removing a factor equal to 1.
$$
\frac{x^{2}-25}{x^{2}-10 x+25}
$$

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Problem 39

Simplify by removing a factor equal to 1.
$$
\frac{t^{2}-1}{t+1}
$$

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Problem 40

Simplify by removing a factor equal to 1.
$$
\frac{a^{2}-1}{a-1}
$$

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Problem 41

Simplify by removing a factor equal to 1.
$$
\frac{y^{2}+4}{y+2}
$$

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Problem 42

Simplify by removing a factor equal to 1.
$$
\frac{x^{2}+1}{x+1}
$$

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Problem 43

Simplify by removing a factor equal to 1.
$$
\frac{5 x^{2}-20}{10 x^{2}-40}
$$

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Problem 44

Simplify by removing a factor equal to 1.
$$
\frac{6 x^{2}-54}{4 x^{2}-36}
$$

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Problem 45

Simplify by removing a factor equal to 1.
$$
\frac{x-8}{8-x}
$$

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Problem 46

Simplify by removing a factor equal to 1.
$$
\frac{6-x}{x-6}
$$

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Problem 47

Simplify by removing a factor equal to 1.
$$
\frac{2 t-1}{1-4 t^{2}}
$$

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Problem 48

Simplify by removing a factor equal to 1.
$$
\frac{3 a-2}{4-9 a^{2}}
$$

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Problem 49

Simplify by removing a factor equal to 1.
$$
\frac{a^{2}-25}{a^{2}+10 a+25}
$$

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Problem 50

Simplify by removing a factor equal to 1.
$$
\frac{a^{2}-16}{a^{2}-8 a+16}
$$

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Problem 51

Simplify by removing a factor equal to 1.
$$
\frac{7 s^{2}-28 t^{2}}{28 t^{2}-7 s^{2}}
$$

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Problem 52

Simplify by removing a factor equal to 1.
$$
\frac{9 m^{2}-4 n^{2}}{4 n^{2}-9 m^{2}}
$$

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Problem 53

Simplify by removing a factor equal to 1.
$$
\frac{x^{3}-1}{x^{2}-1}
$$

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Problem 54

Simplify by removing a factor equal to 1.
$$
\frac{a^{3}+8}{a^{2}-4}
$$

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Problem 55

Simplify by removing a factor equal to 1.
$$
\frac{3 y^{3}+24}{y^{2}-2 y+4}
$$

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Problem 56

Simplify by removing a factor equal to 1.
$$
\frac{x^{3}-27}{5 x^{2}+15 x+45}
$$

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Problem 57

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
f(x)=\frac{3 x+21}{x^{2}+7 x}
$$

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Problem 58

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
f(x)=\frac{5 x+20}{x^{2}+4 x}
$$

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Problem 59

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
g(x)=\frac{x^{2}-9}{5 x+15}
$$

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Problem 60

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
g(x)=\frac{8 x-16}{x^{2}-4}
$$

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Problem 61

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
h(x)=\frac{4-x}{5 x-20}
$$

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Problem 62

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
h(x)=\frac{7-x}{3 x-21}
$$

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Problem 63

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
f(t)=\frac{t^{2}-9}{t^{2}+4 t+3}
$$

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Problem 64

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
f(t)=\frac{t^{2}-25}{t^{2}-6 t+5}
$$

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Problem 65

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
g(t)=\frac{21-7 t}{3 t-9}
$$

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Problem 66

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
g(t)=\frac{12-6 t}{5 t-10}
$$

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Problem 67

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
h(t)=\frac{t^{2}+5 t+4}{t^{2}-8 t-9}
$$

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Problem 68

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
h(t)=\frac{t^{2}-3 t-4}{t^{2}+9 t+8}
$$

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Problem 69

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
f(x)=\frac{9 x^{2}-4}{3 x-2}
$$

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Problem 70

Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.
$$
f(x)=\frac{4 x^{2}-1}{2 x-1}
$$

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Problem 71

Determine the vertical asymptotes of the graph of each function.
$$
f(x)=\frac{3 x-12}{3 x+15}
$$

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Problem 72

Determine the vertical asymptotes of the graph of each function.
$$
f(x)=\frac{4 x-20}{4 x+12}
$$

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Problem 73

Determine the vertical asymptotes of the graph of each function.
$$
g(x)=\frac{12-6 x}{5 x-10}
$$

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Problem 74

Determine the vertical asymptotes of the graph of each function.
$$
r(x)=\frac{21-7 x}{3 x-9}
$$

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Problem 75

Determine the vertical asymptotes of the graph of each function.
$$
t(x)=\frac{x^{3}+3 x^{2}}{x^{2}+6 x+9}
$$

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Problem 76

Determine the vertical asymptotes of the graph of each function.
$$
g(x)=\frac{x^{2}-4}{2 x^{2}-5 x+2}
$$

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Problem 77

Determine the vertical asymptotes of the graph of each function.
$$
f(x)=\frac{x^{2}-x-6}{x^{2}-6 x+8}
$$

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Problem 78

Determine the vertical asymptotes of the graph of each function.
$$
f(x)=\frac{x^{2}+2 x+1}{x^{2}-2 x+1}
$$

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Problem 79

In Exercises 79–84, match each function with one of the following graphs.
$$
h(x)=\frac{1}{x}
$$
(GRAPH CANT COPY)

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Problem 80

In Exercises 79–84, match each function with one of the following graphs.
$$
q(x)=-\frac{1}{x}
$$
(GRAPH CANT COPY)

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Problem 81

In Exercises 79–84, match each function with one of the following graphs.
In Exercises 79–84, match each function with one of the following graphs.
$$
h(x)=\frac{1}{x}
$$
(GRAPH CANT COPY)
(GRAPH CANT COPY)

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Problem 82

In Exercises 79–84, match each function with one of the following graphs.
$$
g(x)=\frac{x-3}{x+2}
$$
(GRAPH CANT COPY)

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Problem 83

In Exercises 79–84, match each function with one of the following graphs.
$$
r(x)=\frac{4 x-2}{x^{2}-2 x+1}
$$
(GRAPH CANT COPY)

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Problem 84

Explain why the graphs of $f(x)=5 x$ and $g(x)=\frac{5 x^{2}}{x}$ differ

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Problem 85

Explain why the graphs of $f(x)=5 x$ and $g(x)=\frac{5 x^{2}}{x}$ differ.

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Problem 86

If a rational expression is undefined for $x=5$ and $x=-3,$ what is the degree of the denominator? Why?

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Problem 87

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$
-\frac{2}{15} \cdot \frac{10}{7}[1.7]
$$

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Problem 88

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$
\left(\frac{3}{4}\right)\left(\frac{-20}{9}\right)[1.7]
$$

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Problem 89

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$
\frac{7}{10} \div\left(-\frac{8}{15}\right)[1.7]
$$

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Problem 90

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$
\frac{7}{10} \div\left(-\frac{8}{15}\right)[1.7]
$$

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Problem 91

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$
\frac{7}{9}-\frac{2}{3} \cdot \frac{6}{7}[1.8]
$$

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Problem 92

To prepare for Section 7.2, review multiplication and division using fraction notation.
Simplify.
$$
\frac{2}{3}-\left(\frac{3}{4}\right)^{2}
$$

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Problem 93

Keith incorrectly simplifies
$$
\frac{x^{2}+x-2}{x^{2}+3 x+2} \text { as } \frac{x-1}{x+2}
$$
He then checks his simplification by evaluating both expressions for $x=1 .$ Use this situation to explain why evaluating is not a foolproof check.

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Problem 94

How could you convince someone that $a-b$ and $b-a$ are opposites of each other?

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Problem 95

Calculate the slope of the line passing through $(a, f(a))$ and $(a+h, f(a+h))$ for the function $f$ given by $f(x)=x^{2}+5 .$ Be sure your answer is simplified.
(GRAPH CANT COPY)

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Problem 96

Calculate the slope of the line passing through the points $(a, f(a))$ and $(a+h, f(a+h))$ for the function $f$ given by $f(x)=3 x^{2} .$ Be sure your answer is simplified.

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Problem 97

Simplify.
$$
\frac{x^{4}-y^{4}}{(y-x)^{4}}
$$

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Problem 98

Simplify.
$$
\frac{16 y^{4}-x^{4}}{\left(x^{2}+4 y^{2}\right)(x-2 y)}
$$

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Problem 99

Simplify.
$$
\frac{(x-1)\left(x^{4}-1\right)\left(x^{2}-1\right)}{\left(x^{2}+1\right)(x-1)^{2}\left(x^{4}-2 x^{2}+1\right)}
$$

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Problem 100

Simplify.
$$
\frac{\left(t^{4}-1\right)\left(t^{2}-9\right)(t-9)^{2}}{\left(t^{4}-81\right)\left(t^{2}+1\right)(t+1)^{2}}
$$

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Problem 101

Simplify.
$$
\frac{a^{3}-2 a^{2}+2 a-4}{a^{3}-2 a^{2}-3 a+6}
$$

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Problem 102

Simplify.
$$
\frac{x^{3}+x^{2}-y^{3}-y^{2}}{x^{2}-2 x y+y^{2}}
$$

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Problem 103

Simplify.
$$
\frac{(t+2)^{3}\left(t^{2}+2 t+1\right)(t+1)}{(t+1)^{3}\left(t^{2}+4 t+4\right)(t+2)}
$$

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Problem 104

Simplify.
$$
\frac{\left(x^{2}-y^{2}\right)\left(x^{2}-2 x y+y^{2}\right)}{(x+y)^{2}\left(x^{2}-4 x y-5 y^{2}\right)}
$$

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Problem 105

Determine the domain and the range of each function from its graph.
(GRAPH CANT COPY)

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Problem 106

Determine the domain and the range of each function from its graph.
(GRAPH CANT COPY)

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Problem 107

Determine the domain and the range of each function from its graph.
(GRAPH CANT COPY)

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Problem 108

Graph the function given by
$$
f(x)=\frac{x^{2}-9}{x-3}
$$
(Hint. Determine the domain of $f$ and simplify.)

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Problem 109

Select any number $x$, multiply by $2,$ add $5,$ multiply by $5,$ subtract $25,$ and divide by $10 .$ What do you get? Explain how this procedure can be used for a number trick.

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