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# College Algebra 11th

## 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 $x .$
$$y=\sqrt{x}$$

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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.
$$f(x)=\sqrt{5 x-4}$$

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### Problem 68

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

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### Problem 69

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

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### Problem 70

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

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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 135

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

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