College Algebra

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

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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Problem 91

Constructing Functions Express the gross salary $G$ of a person who earns $\$ 10$ per hour as a function of the number $x$ of hours worked.

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Problem 92

Constructing Functions Tiffany, 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 93

Population as a Function of Age The function
$$
P(a)=0.015 a^{2}-4.962 a+290.580
$$
represents the population $P$ (in millions) of Americans that are $a$ years of age or older.
(a) Identify the dependent and independent variables.
(b) Evaluate $P(20) .$ Provide a verbal explanation of the meaning of $P(20)$
(c) Evaluate $P(0)$. Provide a verbal explanation of the meaning of $P(0)$

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Problem 94

Number of Rooms The function
$$
N(r)=-1.44 r^{2}+14.52 r-14.96
$$
represents the number $N$ of housing units (in millions) that have $r$ rooms, where $r$ is an integer and $2 \leq r \leq 9$
(a) Identify the dependent and independent variables.
(b) Evaluate $N(3) .$ Provide a verbal explanation of the meaning of $N(3)$

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Problem 95

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? $x=1.1$ seconds? $x=1.2$ seconds? $x=1.3$ 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 96

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? $x=1.1$ seconds? $x=1.2$ 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 97

Cost of Trans-Atlantic 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 98

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
(Image can't copy)

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Problem 99

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 100

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 101

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 $x .$ Determine a function $H$ that represents total health care expenditures in year $x .$

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Problem 102

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 $x .$ Determine a function $N$ that represents the individual's net income (income after taxes) in year $x$

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Problem 103

Profit Function Suppose that the revenue $R,$ in dollars, from selling $x$ cell phones, in hundreds, is $R(x)=-1.2 x^{2}+220 x$ The cost $C,$ in dollars, of selling $x$ cell phones 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 cell phones are sold.
(c) Interpret $P(15)$

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Problem 104

Profit Function Suppose that the revenue $R,$ in dollars, from selling $x$ clocks is $R(x)=30 x$. The cost $C$, in dollars, of selling $x$ clocks is $C(x)=0.1 x^{2}+7 x+400$
(a) Find the profit function, $P(x)=R(x)-C(x)$
(b) Find the profit if $x=30$ clocks are sold.
(c) Interpret $P(30)$

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Problem 105

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 106

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

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Problem 107

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

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Problem 108

Find a function $H$ that multiplies a number $x$ by $3,$ then subtracts the cube of $x$ and divides the result by your age.

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