Unit 5 Exam Review - Additional Problems 32 Polar Equations Pdf

Effort It

1.1 Functions and Function Notation

1 .

1. ⓐ yes
2. ⓑ yeah. (Notation: If two players had been tied for, say, fourth identify, then the name would non take been a function of rank.)

6 .

$y=f\left(x\right)=\frac{\sqrt[\mathrm{three}]{x}}{2}$

9 .

1. ⓐ yes, because each bank business relationship has a single residual at whatever given fourth dimension
2. ⓑ no, because several depository financial institution account numbers may have the same residue
3. ⓒ no, because the same output may correspond to more than one input.

10 .

1. ⓐ Aye, letter grade is a function of percent form;
2. ⓑ No, information technology is not ane-to-one. There are 100 unlike percent numbers we could get simply only about 5 possible letter grades, and so there cannot be only 1 percent number that corresponds to each alphabetic character form.

12 .

No, considering information technology does not laissez passer the horizontal line test.

one.2 Domain and Range

1 .

$\{-\mathrm{five},0,5,10,15\}$

iii .

$\left(-\infty ,\frac{\mathrm{one}}{2}\right)\cup \left(\frac{1}{2},\infty \right)$

4 .

$\left[-\frac{5}{2},\infty \right)$

five .

1. ⓐ values that are less than or equal to –two, or values that are greater than or equal to –1 and less than 3;
2. ⓑ $\left\{x|x\le -\mathrm{ii}\text{or}-1\le \mathrm{10}<3\right\}$ ;
3. ⓒ $(-\infty ,-2]\cup [-1,3)$

half dozen .

domain =[1950,2002] range = [47,000,000,89,000,000]

7 .

domain: $(-\infty ,\mathrm{ii}];$ range: $(-\infty ,0]$

1.3 Rates of Change and Behavior of Graphs

1 .

$\frac{\$2.84-\$2.31}{\mathrm{five}\text{ years}}=\frac{\$0.53}{5\text{ years}}=\$0.106$ per year.

4 .

The local maximum appears to occur at $(-1,28),$ and the local minimum occurs at $(5,-80).$ The function is increasing on $(-\infty ,-1)\cup (\mathrm{five},\infty )$ and decreasing on $(-\mathrm{ane},5).$

one.4 Composition of Functions

1 .

$\begin{array}{l}\left(fg\right)\left(x\right)=f\left(\mathrm{ten}\right)\mathrm{yard}\left(\mathrm{ten}\right)=\left(x-\mathrm{ane}\right)\left({\mathrm{10}}^2-1\right)={\mathrm{ten}}^3-x^2-x+\mathrm{one}\\ \left(f-g\right)\left(x\right)=f\left(x\right)-\mathrm{thou}\left(x\right)=\left(x-1\right)-\left({\mathrm{ten}}^2-1\right)=x-x^2\end{array}$

No, the functions are not the same.

2 .

A gravitational forcefulness is all the same a force, and so $a\left(\mathrm{Thou}(r)\right)$ makes sense as the dispatch of a planet at a distance r from the Dominicus (due to gravity), but $M\left(a(F)\right)$ does non make sense.

3 .

$f(\mathrm{1000}(1))=f(3)=3$ and $g(f(4))=g(1)=3$

4 .

$g(f(2))=\mathrm{chiliad}(5)=3$

six .

$\left[-4,0\right)\cup \left(0,\infty \right)$

7 .

Possible answer:

$\mathrm{chiliad}\left(x\right)=\sqrt{4+x^{\mathrm{ii}}}$
$h\left(\mathrm{10}\right)=\frac{4}{\mathrm{iii}-x}$
$f=h\circ g$

1.five Transformation of Functions

1 .

$$b(t)=h(t)+10=-4.9t^{\mathrm{two}}+30t+10$$

ii .

The graphs of $f(\mathrm{10})$ and $g(x)$ are shown below. The transformation is a horizontal shift. The function is shifted to the left by two units.

4 .

$g\left(\mathrm{10}\right)=\frac{1}{x-1}+1$

6 .

1. ⓐ

   $m(\mathrm{10})=-f(x)$

   $x$	-ii	0	two	iv
   $g(x)$	$-5$	$-10$	$-15$	$-20$

2. ⓑ

   $h(x)=f(-x)$

   $\mathrm{ten}$	-2	0	2	4
   $h(\mathrm{10})$	15	10	5	unknown

vii .

Notice: $g(x)=f(-x)$ looks the same as $f(x)$ .

9 .

$\mathrm{ten}$	2	4	6	8
$\mathrm{thousand}(x)$	9	12	fifteen	0

11 .

$g(x)=f\left(\frac{1}{3}x\right)$ so using the foursquare root role we get $g(x)=\sqrt{\frac{\mathrm{i}}{3}x}$

1.6 Absolute Value Functions

2 .

using the variable $p$ for passing, $\left|p-80\right|\le 20$

3 .

$f(x)=-\left|x+2\right|+3$

5 .

$f(0)=\mathrm{ane},$ so the graph intersects the vertical axis at $(0,1).$ $f(x)=0$ when $x=-5$ and $\mathrm{10}=1$ so the graph intersects the horizontal axis at $(-5,0)$ and $(1,0).$

7 .

$k\le \mathrm{i}$ or $k\ge 7;$ in interval notation, this would be $(-\infty ,1]\cup [\mathrm{vii},\infty )$

1.7 Changed Functions

4 .

The domain of function $f^{-\mathrm{one}}$ is $(-\infty \text{,}-\mathrm{two})$ and the range of part $f^{-\mathrm{one}}$ is $(1,\infty ).$

v .

1. $f(60)=\mathrm{fifty.}$ In 60 minutes, 50 miles are traveled.
2. $f^{-1}(\mathrm{lx})=70.$ To travel threescore miles, it will have seventy minutes.

8 .

$f^{-1}(x)={\left(2-\mathrm{ten}\right)}^2;\text{domain}\text{of}f:\left[0,\infty \right);\text{domain}\text{of}{f}^{-1}:\left(-\infty ,2\right]$

1.i Section Exercises

1 .

A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs accept the same offset coordinate.

3 .

When a vertical line intersects the graph of a relation more than than one time, that indicates that for that input there is more than than 1 output. At any particular input value, there tin be only one output if the relation is to exist a part.

5 .

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to just i input.

27 .

$f(-3)=-\mathrm{eleven};$
$f(\mathrm{two})=-\mathrm{one};$
$f(-a)=-2a-5;$
$-f(a)=-\mathrm{two}a+5;$
$f(a+h)=\mathrm{ii}a+2h-5$

29 .

$f(-\mathrm{three})=\sqrt{5}+5;$
$f(2)=5;$
$f(-a)=\sqrt{\mathrm{two}+a}+5;$
$-f(a)=-\sqrt{2-a}-5;$
$f(a+h)=\sqrt{\mathrm{ii}-a-h}+5$

31 .

$f(-3)=\mathrm{ii};$ $f(2)=\mathrm{ane}-\mathrm{iii}=-\mathrm{two};$
$f(-a)=|-a-1|-|-a+1|;$
$-f(a)=-|a-1|+|a+1|;$
$f(a+h)=|a+h-1|-|a+h+\mathrm{ane}|$

33 .

$\frac{g(\mathrm{ten})-\mathrm{one\; thousand}(a)}{\mathrm{ten}-a}=x+a+2,x\ne a$

35 .

1. ⓐ $f(-\mathrm{two})=\mathrm{fourteen};$
2. ⓑ $x=3$

37 .

1. ⓐ $f(5)=\mathrm{ten};$
2. ⓑ $x=-1$ or $x=\mathrm{iv}$

39 .

1. ⓐ $f(t)=6-\frac{\mathrm{two}}{\mathrm{iii}}t;$
2. ⓑ $f(-3)=8;$
3. ⓒ $t=\mathrm{six}$

53 .

1. ⓐ $f(0)=1;$
2. ⓑ $f(x)=-3,x=-2$ or $x=2$

55 .

not a function and then information technology is also non a one-to-1 role

59 .

role, but non ane-to-one

67 .

$f(x)=1,\mathrm{10}=2$

69 .

$\begin{array}{ccccc}f(-\mathrm{ii})=14;& f(-1)=\mathrm{xi};& f(0)=8;& f(\mathrm{ane})=5;& f(2)=\mathrm{two}\end{array}$

71 .

$\begin{array}{ccccc}f(-2)=4;& f(-\mathrm{ane})=4.414;& f(0)=4.732;& f(\mathrm{one})=5;& f(2)=5.236\end{array}$

73 .

$\begin{array}{ccccc}f(-\mathrm{two})=\frac{1}{9};& f(-\mathrm{one})=\frac{1}{\mathrm{three}};& f(0)=\mathrm{ane};& f(1)=3;& f(2)=9\end{array}$

77 .

$[0,\text{ 100}]$

79 .

$[-0.001,\text{ 0}\text{.001}]$

81 .

$[-1,000,000,\text{ one,000,000}]$

83 .

$[0,\text{ 10}]$

85 .

$[\mathrm{-0.1},\text{0.1}]$

87 .

$[-100,\text{ 100}]$

89 .

1. ⓐ $g(5000)=\mathrm{fifty};$
2. ⓑ The number of cubic yards of dirt required for a garden of 100 square anxiety is ane.

91 .

1. ⓐ The acme of a rocket in a higher place basis after one second is 200 ft.
2. ⓑ the meridian of a rocket above footing after 2 seconds is 350 ft.

one.ii Section Exercises

1 .

The domain of a office depends upon what values of the independent variable make the office undefined or imaginary.

3 .

There is no restriction on $x$ for $f(x)=\sqrt[\mathrm{iii}]{x}$ because you can have the cube root of any real number. And then the domain is all real numbers, $(-\infty ,\infty ).$ When dealing with the set of real numbers, you cannot take the square root of negative numbers. Then $\mathrm{ten}$ -values are restricted for $f(x)=\sqrt[]{\mathrm{ten}}$ to nonnegative numbers and the domain is $[0,\infty ).$

5 .

Graph each formula of the piecewise function over its corresponding domain. Utilize the same scale for the $\mathrm{10}$ -centrality and $y$ -axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Employ an arrow to signal $-\infty$ or $\infty .$ Combine the graphs to discover the graph of the piecewise part.

xv .

$(-\infty ,-\frac{1}{\mathrm{two}})\cup (-\frac{1}{2},\infty )$

17 .

$(-\infty ,-11)\cup (-11,\mathrm{ii})\cup (\mathrm{ii},\infty )$

xix .

$(-\infty ,-\mathrm{iii})\cup (-3,5)\cup (5,\infty )$

25 .

$\left(-\infty ,-9\right)\cup \left(-9,9\right)\cup \left(9,\infty \right)$

27 .

domain: $(2,8],$ range $[6,\mathrm{eight})$

29 .

domain: $[-4,\text{ iv],}$ range: $[0,\text{ 2]}$

31 .

domain: $[-5,3),$ range: $\left[0,\mathrm{ii}\right]$

33 .

domain: $(-\infty ,1],$ range: $[0,\infty )$

35 .

domain: $\left[-6,-\frac{1}{6}\right]\cup \left[\frac{1}{6},\mathrm{six}\right];$ range: $\left[-6,-\frac{1}{\mathrm{half-dozen}}\right]\cup \left[\frac{1}{\mathrm{six}},\mathrm{vi}\right]$

37 .

domain: $[-3,\infty );$ range: $[0,\infty )$

39 .

domain: $(-\infty ,\infty )$

41 .

domain: $(-\infty ,\infty )$

43 .

domain: $(-\infty ,\infty )$

45 .

domain: $(-\infty ,\infty )$

47 .

$\begin{array}{cccc}f(-\mathrm{three})=1;& f(-2)=0;& f(-1)=0;& f(0)=0\end{array}$

49 .

$\begin{array}{cccc}f(-1)=-4;& f(0)=6;& f(2)=20;& f(\mathrm{iv})=34\end{array}$

51 .

$\begin{array}{cccc}f(-1)=-\mathrm{v};& f(0)=\mathrm{three};& f(\mathrm{ii})=3;& f(\mathrm{iv})=16\end{array}$

53 .

domain: $(-\infty ,1)\cup (1,\infty )$

55 .

window: $[-\mathrm{0.five},-0.1];$ range: $[4,100]$

window: $[0.1,\mathrm{0.v}];$ range: $[4,100]$

59 .

Many answers. 1 function is $f(x)=\frac{1}{\sqrt{\mathrm{ten}-2}}.$

one.3 Section Exercises

1 .

Yeah, the average rate of change of all linear functions is constant.

3 .

The absolute maximum and minimum relate to the unabridged graph, whereas the local extrema relate only to a specific region around an open up interval.

xi .

$\frac{-1}{\mathrm{xiii}\left(\mathrm{thirteen}+h\right)}$

13 .

$3h^2+\mathrm{nine}h+9$

xix .

increasing on $\left(-\infty ,-2.5\right)\cup \left(\mathrm{ane},\infty \right),$ decreasing on $(-\mathrm{two.5},\mathrm{i})$

21 .

increasing on $\left(-\infty ,\mathrm{i}\right)\cup \left(3,4\right),$ decreasing on $\left(\mathrm{ane},3\right)\cup \left(\mathrm{iv},\infty \right)$

23 .

local maximum: $(-\mathrm{three},50),$ local minimum: $(\mathrm{three},-\mathrm{l})$

25 .

absolute maximum at approximately $(7,150),$ absolute minimum at approximately $(\mathrm{-vii.5},\mathrm{-220})$

35 .

Local minimum at $(3,-22),$ decreasing on $(-\infty ,\mathrm{three}),$ increasing on $(3,\infty )$

37 .

Local minimum at $(-2,-2),$ decreasing on $(-3,-2),$ increasing on $(-\mathrm{two},\infty )$

39 .

Local maximum at $(-\mathrm{0.v},6),$ local minima at $(-3.25,-47)$ and $(\mathrm{ii.i},-32),$ decreasing on $(-\infty ,-3.25)$ and $(-\mathrm{0.v},\mathrm{ii.1}),$ increasing on $(-3.25,-\mathrm{0.five})$ and $(2.1,\infty )$

45 .

2.seven gallons per minute

47 .

approximately –0.6 milligrams per day

1.4 Section Exercises

ane .

Notice the numbers that brand the function in the denominator $\mathrm{thou}$ equal to zip, and check for any other domain restrictions on $f$ and $k,$ such as an fifty-fifty-indexed root or zeros in the denominator.

3 .

Yes. Sample answer: Let $f(x)=x+1\text{ and}\mathrm{yard}(x)=x-1.$ Then $f(g(x))=f(x-1)=(\mathrm{ten}-1)+1=x$ and $g(f(\mathrm{10}))=k(x+1)=(\mathrm{10}+\mathrm{i})-1=x.$ So $f\circ g=m\circ f.$

5 .

$(f+\mathrm{grand})\left(\mathrm{10}\right)=2x+6,$ domain: $(-\infty ,\infty )$

$(f-g)\left(x\right)=2x^2+\mathrm{ii}x-6,$ domain: $(-\infty ,\infty )$

$(fg)\left(x\right)=-x^4-2x^3+6{\mathrm{10}}^2+12x,$ domain: $(-\infty ,\infty )$

$\left(\frac{f}{g}\right)\left(x\right)=\frac{x^2+2\mathrm{10}}{\mathrm{half-dozen}-x^2},$ domain: $(-\infty ,-\sqrt{6})\cup (-\sqrt{\mathrm{vi}},\sqrt{6})\cup (\sqrt{6},\infty )$

7 .

$(f+\mathrm{thousand})\left(x\right)=\frac{4x^3+8{\mathrm{ten}}^2+1}{2\mathrm{ten}},$ domain: $(-\infty ,0)\cup (0,\infty )$

$(f-\mathrm{1000})\left(\mathrm{10}\right)=\frac{4x^{\mathrm{three}}+8{\mathrm{10}}^{\mathrm{ii}}-1}{2x},$ domain: $(-\infty ,0)\cup (0,\infty )$

$(fg)\left(x\right)=x+\mathrm{two},$ domain: $(-\infty ,0)\cup (0,\infty )$

$\left(\frac{f}{\mathrm{thousand}}\right)\left(x\right)=\mathrm{four}{x}^3+8x^2,$ domain: $(-\infty ,0)\cup (0,\infty )$

9 .

$(f+\mathrm{1000})(x)=\mathrm{three}{\mathrm{ten}}^2+\sqrt{x-5},$ domain: $[5,\infty )$

$(f-\mathrm{thousand})(\mathrm{ten})=\mathrm{three}{\mathrm{ten}}^2-\sqrt{\mathrm{ten}-\mathrm{v}},$ domain: $[\mathrm{five},\infty )$

$(fg)(x)=\mathrm{three}{x}^2\sqrt{\mathrm{10}-\mathrm{v}},$ domain: $[5,\infty )$

$\left(\frac{f}{g}\right)(x)=\frac{\mathrm{iii}{\mathrm{10}}^2}{\sqrt{x-5}},$ domain: $(5,\infty )$

xi .

1. ⓐ 3
2. ⓑ $f\left(g\left(x\right)\right)=\mathrm{two}{\left(\mathrm{three}x-\mathrm{five}\right)}^2+1;$
3. ⓒ $\mathrm{thousand}\left(f)\left(x\right)\right)=6x^{\mathrm{ii}}-2;$
4. ⓓ $\left(\mathrm{thou}\circ \mathrm{grand}\right)(x)=3(\mathrm{iii}x-\mathrm{v})-\mathrm{five}=9x-20;$
5. ⓔ $\left(f\circ f\right)\left(-\mathrm{two}\right)=163$

13 .

$f(g(x))=\sqrt{x^2+3}+2,g(f(x))=x+4\sqrt{x}+\mathrm{vii}$

xv .

$f(\mathrm{thou}(x))=\sqrt[3]{\frac{\mathrm{ten}+1}{x^3}}=\frac{\sqrt[3]{\mathrm{10}+\mathrm{ane}}}{\mathrm{ten}},k(f(\mathrm{ten}))=\frac{\sqrt[3]{x}+\mathrm{ane}}{\mathrm{10}}$

17 .

$\left(f\circ g\right)(x)=\frac{1}{\frac{2}{\mathrm{ten}}+4-4}=\frac{\mathrm{ten}}{\mathrm{two}},\left(g\circ f\right)(x)=2x-4$

nineteen .

$f(\mathrm{one\; thousand}(h(x)))={\left(\frac{\mathrm{one}}{x+3}\right)}^2+1$

21 .

• ⓐ Text $(g\circ f)(x)=-\frac{3}{\sqrt{\mathrm{two}-4x}};$
• ⓑ $\left(-\infty ,\frac{1}{2}\right)$

23 .

1. ⓐ $(0,2)\cup (2,\infty );$
2. ⓑ $(-\infty ,-2)\cup (2,\infty );$ c. $(0,\infty )$

27 .

sample: $\begin{array}{l}f(x)={\mathrm{10}}^{\mathrm{iii}}\\ \mathrm{1000}(x)=x-5\end{array}$

29 .

sample: $\begin{array}{l}f(x)=\frac{\mathrm{four}}{x}\hfill \\ \mathrm{thou}(x)={(\mathrm{10}+\mathrm{two})}^{\mathrm{ii}}\hfill \end{array}$

31 .

sample: $\begin{array}{l}f(x)=\sqrt[3]{x}\\ g(x)=\frac{1}{2x-3}\end{array}$

33 .

sample: $\begin{array}{l}f(x)=\sqrt[4]{\mathrm{10}}\\ \mathrm{yard}(x)=\frac{\mathrm{iii}\mathrm{10}-2}{x+5}\end{array}$

35 .

sample: $f(x)=\sqrt{x}$
$g(\mathrm{ten})=\mathrm{two}\mathrm{ten}+\mathrm{half-dozen}$

37 .

sample: $f(x)=\sqrt[\mathrm{three}]{\mathrm{10}}$
$g(x)=(\mathrm{ten}-\mathrm{one})$

39 .

sample: $f(\mathrm{10})=x^{\mathrm{iii}}$
$\mathrm{1000}(x)=\frac{1}{\mathrm{ten}-2}$

41 .

sample: $f(x)=\sqrt{x}$
$g(x)=\frac{2x-\mathrm{i}}{3x+4}$

73 .

$f(g(0))=27,g\left(f(0)\right)=-94$

75 .

$f(g(0))=\frac{1}{5},\mathrm{chiliad}(f(0))=5$

77 .

$\mathrm{xviii}{\mathrm{ten}}^2+60\mathrm{10}+51$

79 .

$\mathrm{thou}\circ g(x)=9\mathrm{10}+\mathrm{twenty}$

87 .

$(f\circ g)(\mathrm{six})=6$ ; $(k\circ f)(6)=6$

89 .

$(f\circ g)(11)=\mathrm{xi},(\mathrm{yard}\circ f)(\mathrm{eleven})=11$

93 .

$A(t)=\pi {\left(25\sqrt{t+2}\right)}^{\mathrm{ii}}$ and $A(2)=\pi {\left(25\sqrt{\mathrm{iv}}\right)}^2=2500\pi$ square inches

95 .

$A(\mathrm{five})=\pi {\left(2(5)+\mathrm{one}\right)}^{\mathrm{two}}=121\pi$ square units

97 .

• ⓐ $\mathrm{Due\; north}(T(t))=23{(5t+\mathrm{ane.5})}^2-56(\mathrm{five}t+1.5)+1;$
• ⓑ 3.38 hours

1.v Section Exercises

i .

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

iii .

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and i is multiplied by the output.

5 .

For a function $f,$ substitute $(-\mathrm{10})$ for $(x)$ in $f(x).$ Simplify. If the resulting function is the same as the original role, $f(-x)=f(\mathrm{ten}),$ then the function is even. If the resulting role is the reverse of the original role, $f(-x)=-f(x),$ and then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

seven .

$\mathrm{1000}(x)=|x-1|-3$

9 .

$\mathrm{thou}(\mathrm{10})=\frac{\mathrm{ane}}{{(\mathrm{10}+4)}^{\mathrm{two}}}+2$

11 .

The graph of $f(x+43)$ is a horizontal shift to the left 43 units of the graph of $f.$

13 .

The graph of $f(\mathrm{10}-4)$ is a horizontal shift to the right 4 units of the graph of $f.$

15 .

The graph of $f(\mathrm{ten})+8$ is a vertical shift up 8 units of the graph of $f.$

17 .

The graph of $f(x)-7$ is a vertical shift down 7 units of the graph of $f.$

19 .

The graph of $f(x+4)-\mathrm{i}$ is a horizontal shift to the left four units and a vertical shift down ane unit of the graph of $f.$

21 .

decreasing on $(-\infty ,-\mathrm{three})$ and increasing on $(-3,\infty )$

23 .

decreasing on $[0,\infty )$

31 .

$\mathrm{1000}(x)=f(\mathrm{ten}-\mathrm{one}),h(\mathrm{10})=f(\mathrm{10})+\mathrm{ane}$

33 .

$f(\mathrm{10})=|x-3|-2$

35 .

$f(x)=\sqrt{x+3}-1$

37 .

$f(x)={(x-2)}^2$

39 .

$f(x)=|x+\mathrm{three}|-2$

43 .

$f(x)=-{(x+1)}^2+\mathrm{two}$

45 .

$f(x)=\sqrt{-x}+1$

53 .

The graph of $g$ is a vertical reflection (across the $x$ -axis) of the graph of $f.$

55 .

The graph of $g$ is a vertical stretch past a factor of 4 of the graph of $f.$

57 .

The graph of $g$ is a horizontal pinch past a factor of $\frac{1}{\mathrm{five}}$ of the graph of $f.$

59 .

The graph of $g$ is a horizontal stretch by a factor of 3 of the graph of $f.$

61 .

The graph of $g$ is a horizontal reflection across the $y$ -axis and a vertical stretch by a factor of three of the graph of $f.$

63 .

$k(x)=|-\mathrm{four}x|$

65 .

$\mathrm{1000}(x)=\frac{1}{3{(\mathrm{10}+2)}^2}-3$

67 .

$\mathrm{grand}(x)=\frac{\mathrm{i}}{2}{(x-5)}^2+\mathrm{i}$

69 .

The graph of the function $f(\mathrm{ten})=x^{\mathrm{two}}$ is shifted to the left 1 unit, stretched vertically by a cistron of 4, and shifted downwards 5 units.

71 .

The graph of $f(x)=\left|x\right|$ is stretched vertically past a factor of ii, shifted horizontally iv units to the right, reflected across the horizontal axis, and then shifted vertically three units up.

73 .

The graph of the office $f(x)={\mathrm{10}}^{\mathrm{three}}$ is compressed vertically by a factor of $\frac{\mathrm{ane}}{2}.$

75 .

The graph of the function is stretched horizontally by a gene of 3 and then shifted vertically downwardly past 3 units.

77 .

The graph of $f(x)=\sqrt{x}$ is shifted correct 4 units and and so reflected across the vertical line $x=4.$

1.vi Section Exercises

one .

Isolate the absolute value term and so that the equation is of the form $\left|A\right|=B.$ Form one equation past setting the expression inside the accented value symbol, $A,$ equal to the expression on the other side of the equation, $B.$ Form a second equation past setting $A$ equal to the opposite of the expression on the other side of the equation, $-B.$ Solve each equation for the variable.

3 .

The graph of the absolute value function does not cross the $x$ -axis, then the graph is either completely above or completely below the $x$ -centrality.

5 .

Starting time determine the purlieus points by finding the solution(s) of the equation. Employ the boundary points to form possible solution intervals. Cull a test value in each interval to determine which values satisfy the inequality.

seven .

$\left|x+\mathrm{iv}\right|=\frac{1}{\mathrm{ii}}$

9 .

$|f(x)-\mathrm{viii}|<0.03$

13 .

$\{-\frac{9}{4},\frac{13}{\mathrm{four}}\}$

fifteen .

$\left\{\frac{10}{\mathrm{three}},\frac{20}{3}\right\}$

17 .

$\left\{\frac{\mathrm{eleven}}{\mathrm{v}},\frac{29}{\mathrm{five}}\right\}$

19 .

$\left\{\frac{5}{2},\frac{\mathrm{seven}}{2}\right\}$

23 .

$\left\{-57,27\right\}$

25 .

$\left(0,-8\right);\left(-6,0\right),\left(\mathrm{four},0\right)$

27 .

$\left(0,-7\right);$ no $\mathrm{10}$ -intercepts

29 .

$(-\infty ,-8)\cup (12,\infty )$

33 .

$\left(-\infty ,-\frac{8}{\mathrm{three}}\right]\cup \left[\mathrm{six},\infty \right)$

35 .

$\left(-\infty ,-\frac{8}{3}\right]\cup \left[16,\infty \right)$

53 .

range: $\left[0,\mathrm{xx}\right]$

55 .

$x\text{-}$ intercepts:

59 .

At that place is no solution for $a$ that will keep the function from having a $y$ -intercept. The absolute value function always crosses the $y$ -intercept when $x=0.$

61 .

$\left|p-0.08\right|\le 0.015$

63 .

$\left|x-\mathrm{v.0}\right|\le 0.01$

i.7 Section Exercises

i .

Each output of a function must have exactly ane output for the function to exist 1-to-one. If whatever horizontal line crosses the graph of a function more than once, that means that $y$ -values echo and the function is not ane-to-one. If no horizontal line crosses the graph of the function more than one time, so no $y$ -values echo and the function is one-to-1.

3 .

Yep. For example, $f(x)=\frac{1}{x}$ is its own inverse.

5 .

Given a role $y=f(\mathrm{ten}),$ solve for $\mathrm{ten}$ in terms of $y.$ Interchange the $\mathrm{ten}$ and $y.$ Solve the new equation for $y.$ The expression for $y$ is the inverse, $y=f^{-\mathrm{i}}(x).$

vii .

$f^{-1}(x)=\mathrm{10}-\mathrm{iii}$

9 .

$f^{-\mathrm{ane}}(\mathrm{ten})=\mathrm{ii}-x$

11 .

$f^{-1}(x)=\frac{-\mathrm{two}x}{x-1}$

13 .

domain of $f(\mathrm{10}):[-7,\infty );f^{-\mathrm{ane}}(x)=\sqrt{x}-7$

15 .

domain of $f(x):[0,\infty );f^{-\mathrm{i}}(\mathrm{ten})=\sqrt{\mathrm{ten}+5}$

xvi .

• ⓐ $f(\mathrm{thousand}(x))=x$ and $g(f(x))=x.$
• ⓑ This tells u.s.a. that $f$ and $g$ are inverse functions

17 .

$f(g(x))=x,\mathrm{one\; thousand}(f(\mathrm{ten}))=\mathrm{ten}$

41 .

$x$	i	4	7	12	16
$f^{-1}(x)$	3	six	9	13	14

43 .

$f^{-1}(x)={(1+x)}^{1/3}$

45 .

$f^{-\mathrm{ane}}(x)=\frac{5}{\mathrm{ix}}\left(\mathrm{ten}-32\right).$ Given the Fahrenheit temperature, $x,$ this formula allows yous to calculate the Celsius temperature.

47 .

$t(d)=\frac{d}{50},$ $t(180)=\frac{180}{\mathrm{l}}.$ The time for the car to travel 180 miles is 3.6 hours.

Review Exercises

5 .

$f(-3)=-27;$ $f(\mathrm{two})=-2;$ $f(-a)=-2a^{\mathrm{ii}}-3a;$
$-f(a)=2a^2-3a;$ $f(a+h)=-2a^2+3a-4ah+3h-\mathrm{two}{h}^2$

17 .

$x=-1.8$ or $\text{ or}x=\mathrm{i.8}$

xix .

$\frac{-64+\mathrm{lxxx}a-16a^2}{-\mathrm{ane}+a}=-16a+64$

21 .

$\left(-\infty ,-2\right)\cup \left(-\mathrm{two},\mathrm{six}\right)\cup \left(6,\infty \right)$

27 .

increasing $\left(2,\infty \right);$ decreasing $(-\infty ,2)$

29 .

increasing $\left(-3,1\right);$ constant $(-\infty ,-3)\cup \left(1,\infty \right)$

31 .

local minimum $\left(-\mathrm{ii},-3\right);$ local maximum $\left(1,3\right)$

33 .

Absolute Maximum: x

35 .

$\left(f\circ g\right)(x)=17-18\mathrm{10};\left(g\circ f\right)(\mathrm{10})=-\mathrm{vii}-18\mathrm{10}$

37 .

$\left(f\circ g\right)(\mathrm{10})=\sqrt{\frac{\mathrm{ane}}{\mathrm{10}}+\mathrm{two}};$ $\left(g\circ f\right)(x)=\frac{1}{\sqrt{\mathrm{ten}+\mathrm{ii}}}$

39 .

$(f\circ g)(x)=\frac{\mathrm{ane}+\mathrm{10}}{\mathrm{ane}+\mathrm{iv}x},\mathrm{10}\ne 0,x\ne -\frac{1}{4}$

41 .

$\left(f\circ g\right)(x)=\frac{1}{\sqrt{x}},x>0$

43 .

sample: $\mathrm{1000}(x)=\frac{2x-1}{3x+4};f(x)=\sqrt{x}$

55 .

$f(\mathrm{ten})=\left|\mathrm{ten}-3\right|$

63 .

$f(x)=\frac{1}{2}\left|x+\mathrm{two}\right|+1$

65 .

$f(\mathrm{ten})=-\mathrm{iii}\left|x-3\right|+3$

69 .

$x=-22,\mathrm{10}=\mathrm{fourteen}$

71 .

$\left(-\frac{5}{3},3\right)$

73 .

$f^{-1}(x)=\sqrt{x-\mathrm{i}}$

77 .

The office is one-to-1.

78 .

The function is not one-to-1.

Exercise Test

1 .

The relation is a function.

5 .

The graph is a parabola and the graph fails the horizontal line test.

19 .

$x=-\mathrm{vii}$ and $\mathrm{10}=\mathrm{x}$

21 .

$f^{-1}(x)=\frac{\mathrm{ten}+5}{3}$

23 .

$(-\infty ,-\mathrm{1.ane})\text{ and}(1.1,\infty )$

25 .

$\left(\mathrm{1.i},-\mathrm{0.ix}\right)$

29 .

$f(x)=\{\begin{array}{c}\left|\mathrm{ten}\right|\text{if}\mathrm{10}\le 2\\ \mathrm{three}\text{if}\mathrm{10}>2\end{array}$

35 .

$f^{-1}(x)=-\frac{\mathrm{10}-\mathrm{eleven}}{2}$

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Source: https://openstax.org/books/precalculus/pages/chapter-1

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