# College Algebra

## 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

If $f$ is a function defined by the equation $y=f(x),$ then $x$ is called the ______ variable and $y$ is the ______ variable.

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### Problem 6

The set of all images of the elements in the domain of a function is called the _______

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### Problem 7

If the domain of $f$ is all real numbers in the interval $[0,7]$ and the domain of $g$ is all real numbers in the interval $[-2,5],$ the domain of $f+g$ is all real numbers in the interval ________.

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### Problem 8

The domain of $\frac{f}{g}$ consists of numbers $x$ for which g(x) ______ 0 that are in the domains of both _____ and _____.

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### Problem 9

If $f(x)=x+1$ and $g(x)=x^{3},$ then ______ =$x^{3}-(x+1)$

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### Problem 10

True or False Every relation is a function.

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### Problem 11

True or False The domain of $(f \cdot g)(x)$ consists of the numbers $x$ that are in the domains of both $f$ and $g .$

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### Problem 12

True or False The independent variable is sometimes referred to as the argument of the function.

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### Problem 13

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

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### Problem 14

True or False The domain of the function $f(x)=\frac{x^{2}-4}{x}$ is $\{x | x \neq \pm 2\}$

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### Problem 15

Determine whether each relation represents a function. For each function, state the domain and range.
(function can't copy)

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### Problem 16

Determine whether each relation represents a function. For each function, state the domain and range.
(function can't copy)

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### Problem 17

Determine whether each relation represents a function. For each function, state the domain and range.
(function can't copy)

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### Problem 18

Determine whether each relation represents a function. For each function, state the domain and range.
(function can't copy)

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### Problem 19

Determine whether each relation represents a function. For each function, state the domain and range.
$$\{(2,6),(-3,6),(4,9),(2,10)\}$$

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### Problem 20

Determine whether each relation represents a function. For each function, state the domain and range.
$$\{(-2,5),(-1,3),(3,7),(4,12)\}$$

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### Problem 21

Determine whether each relation represents a function. For each function, state the domain and range.
$$\{(1,3),(2,3),(3,3),(4,3)\}$$

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### Problem 22

Determine whether each relation represents a function. For each function, state the domain and range.
$$\{(0,-2),(1,3),(2,3),(3,7)\}$$

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### Problem 23

Determine whether each relation represents a function. For each function, state the domain and range.
$$\{(-2,4),(-2,6),(0,3),(3,7)\}$$

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### Problem 24

Determine whether each relation represents a function. For each function, state the domain and range.
$$\{(-4,4),(-3,3),(-2,2),(-1,1),(-4,0)\}$$

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### Problem 25

Determine whether each relation represents a function. For each function, state the domain and range.
$$\{(-2,4),(-1,1),(0,0),(1,1)\}$$

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### Problem 26

Determine whether each relation represents a function. For each function, state the domain and range.
$$\{(-2,16),(-1,4),(0,3),(1,4)\}$$

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### Problem 27

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

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### Problem 28

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

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### Problem 29

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

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### Problem 30

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

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### Problem 31

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

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### Problem 32

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

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### Problem 33

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

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### Problem 34

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

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### Problem 35

Determine whether the equation defines y as a function of x.
$$y=2 x^{2}-3 x+4$$

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### Problem 36

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

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### Problem 37

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

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### Problem 38

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

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### Problem 39

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 40

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 41

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 42

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 43

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

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

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

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### Problem 47

Find the domain of each function.
$$f(x)=-5 x+4$$

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### Problem 48

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

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### Problem 49

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

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### Problem 50

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

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### Problem 51

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

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### Problem 52

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

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### Problem 53

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

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### Problem 54

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

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### Problem 55

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

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### Problem 56

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

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### Problem 57

Find the domain of each function.
$$f(x)=\frac{4}{\sqrt{x-9}}$$

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### Problem 58

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

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### Problem 59

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

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### Problem 60

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

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### Problem 61

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

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### Problem 62

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

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### Problem 63

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=3 x+4 ; \quad g(x)=2 x-3$$

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### Problem 64

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=2 x+1 ; \quad g(x)=3 x-2$$

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### Problem 65

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=x-1 ; \quad g(x)=2 x^{2}$$

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### Problem 66

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=2 x^{2}+3 ; \quad g(x)=4 x^{3}+1$$

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### Problem 67

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=\sqrt{x} ; \quad g(x)=3 x-5$$

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### Problem 68

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=|x| ; \quad g(x)=x$$

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### Problem 69

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=|x| ; \quad g(x)=x$$

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### Problem 70

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=\sqrt{x-1} ; \quad g(x)=\sqrt{4-x}$$

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### Problem 71

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=\frac{2 x+3}{3 x-2} ; \quad g(x)=\frac{4 x}{3 x-2}$$

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### Problem 72

For the given functions $f$ and $g,$ find the following. For parts ( $a$ )-( $d$ ), also find the domain.
(a)$(f+g)(x)$
(b) $(f-g)(x)$
(c) $(f \cdot g)(x)$
(d)$\left(\frac{f}{g}\right)(x)$
(e)$(f+g)(3)$
(f)$(f-g)(4)$
(g) $(f \cdot g)(2)$
( h)$\left(\frac{f}{g}\right)(1)$
$$f(x)=\sqrt{x+1} ; \quad g(x)=\frac{2}{x}$$

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### Problem 73

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

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### Problem 74

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

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### Problem 75

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 76

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 77

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 78

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 79

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 80

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 81

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.
$$f(x)=\sqrt{x}$$
[Hint: Rationalize the numerator.

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### Problem 82

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 83

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 84

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

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### Problem 85

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 86

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 87

If $f(x)=\frac{2 x-A}{x-3}$ and $f(4)=0,$ what is the value of $A ?$ Where is $f$ not defined?

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### Problem 88

If $f(x)=\frac{x-B}{x-A}, f(2)=0$ and $f(1)$ is undefined, what are the values of $A$ and $B ?$

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### Problem 89

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 90

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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