Precalculus/trig Name Unit 1 Algebra Ii Review – Section 3 Worksheet

Try It

1.one Functions and Function Notation

1 .

1. ⓐ aye
2. ⓑ yes. (Note: If 2 players had been tied for, say, 4th place, then the name would not accept been a function of rank.)

6 .

$y=f\left(x\right)=\frac{\sqrt[3]{x}}{\mathrm{two}}$

9 .

1. ⓐ yes, because each bank account has a single balance at any given time
2. ⓑ no, because several bank account numbers may have the aforementioned balance
3. ⓒ no, because the aforementioned output may represent to more than one input.

ten .

1. ⓐ Yes, letter grade is a function of percent course;
2. ⓑ No, information technology is non i-to-one. There are 100 different percent numbers we could go merely only about 5 possible letter grades, so there cannot be merely i percentage number that corresponds to each letter of the alphabet grade.

12 .

No, because it does non pass the horizontal line examination.

1.two Domain and Range

1 .

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

3 .

$\left(-\infty ,\frac{1}{\mathrm{ii}}\right)\cup \left(\frac{1}{\mathrm{two}},\infty \right)$

4 .

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

v .

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

6 .

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

vii .

domain: $(-\infty ,2];$ range: $(-\infty ,0]$

ane.3 Rates of Change and Beliefs of Graphs

i .

$\frac{\$\mathrm{two.84}-\$2.31}{\mathrm{five}\text{ years}}=\frac{\$0.53}{\mathrm{five}\text{ years}}=\$0.106$ per year.

4 .

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

one.4 Composition of Functions

i .

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

No, the functions are non the same.

2 .

A gravitational force is nonetheless a forcefulness, and so $a\left(G(r)\right)$ makes sense equally the acceleration of a planet at a distance r from the Dominicus (due to gravity), but $G\left(a(F)\right)$ does not make sense.

3 .

$f(g(\mathrm{i}))=f(\mathrm{three})=3$ and $k(f(\mathrm{iv}))=\mathrm{chiliad}(\mathrm{i})=3$

4 .

$\mathrm{chiliad}(f(2))=g(\mathrm{five})=\mathrm{three}$

6 .

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

7 .

Possible reply:

$g\left(\mathrm{10}\right)=\sqrt{4+x^2}$
$h\left(x\right)=\frac{4}{3-x}$
$f=h\circ g$

1.5 Transformation of Functions

i .

$$b(t)=h(t)+\mathrm{x}=-\mathrm{iv.nine}{t}^{\mathrm{two}}+\mathrm{xxx}t+10$$

ii .

The graphs of $f(x)$ and $\mathrm{thousand}(x)$ are shown beneath. The transformation is a horizontal shift. The office is shifted to the left past ii units.

four .

$k\left(x\right)=\frac{1}{x-1}+\mathrm{ane}$

half dozen .

1. ⓐ

   $g(x)=-f(x)$

   $\mathrm{ten}$	-2	0	2	4
   $\mathrm{one\; thousand}(x)$	$-5$	$-\mathrm{x}$	$-15$	$-20$

2. ⓑ

   $h(\mathrm{ten})=f(-\mathrm{ten})$

   $x$	-2	0	2	four
   $h(x)$	15	10	5	unknown

vii .

Notice: $g(\mathrm{ten})=f(-x)$ looks the aforementioned as $f(x)$ .

nine .

$x$	2	4	6	viii
$g(x)$	9	12	15	0

xi .

$g(x)=f\left(\frac{1}{3}x\right)$ so using the square root function we go $\mathrm{one\; thousand}(\mathrm{ten})=\sqrt{\frac{1}{3}x}$

1.half-dozen Accented Value Functions

2 .

using the variable $p$ for passing, $\left|p-\mathrm{lxxx}\right|\le \mathrm{xx}$

3 .

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

five .

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

vii .

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

1.vii Changed Functions

four .

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

v .

1. $f(60)=50.$ In 60 minutes, l miles are traveled.
2. $f^{-1}(\mathrm{lx})=70.$ To travel threescore miles, information technology will take lxx minutes.

viii .

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

1.1 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 aforementioned start coordinate.

3 .

When a vertical line intersects the graph of a relation more than than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

5 .

When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than 1 input. A role is i-to-one if each output corresponds to merely one input.

27 .

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

29 .

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

31 .

$f(-3)=2;$ $f(2)=1-\mathrm{three}=-2;$
$f(-a)=|-a-\mathrm{ane}|-|-a+1|;$
$-f(a)=-|a-\mathrm{ane}|+|a+\mathrm{one}|;$
$f(a+h)=|a+h-\mathrm{one}|-|a+h+1|$

33 .

$\frac{\mathrm{chiliad}(\mathrm{10})-g(a)}{x-a}=x+a+2,x\ne a$

35 .

1. ⓐ $f(-2)=14;$
2. ⓑ $\mathrm{10}=3$

37 .

1. ⓐ $f(5)=10;$
2. ⓑ $\mathrm{10}=-1$ or $\mathrm{ten}=4$

39 .

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

53 .

1. ⓐ $f(0)=1;$
2. ⓑ $f(\mathrm{10})=-3,x=-2$ or $x=2$

55 .

not a function then it is as well not a 1-to-one function

59 .

function, but non ane-to-ane

67 .

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

69 .

$\begin{array}{ccccc}f(-\mathrm{two})=\mathrm{xiv};& f(-\mathrm{ane})=11;& f(0)=\mathrm{viii};& f(1)=5;& f(2)=2\end{array}$

71 .

$\begin{array}{ccccc}f(-2)=4;& f(-1)=\mathrm{four.414};& f(0)=4.732;& f(\mathrm{i})=\mathrm{five};& f(2)=\mathrm{five.236}\end{array}$

73 .

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

77 .

$[0,\text{ 100}]$

79 .

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

81 .

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

83 .

$[0,\text{ 10}]$

85 .

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

87 .

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

89 .

1. ⓐ $g(5000)=50;$
2. ⓑ The number of cubic yards of dirt required for a garden of 100 square anxiety is 1.

91 .

1. ⓐ The height of a rocket above basis after ane second is 200 ft.
2. ⓑ the height of a rocket higher up basis after 2 seconds is 350 ft.

one.2 Section Exercises

1 .

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

iii .

In that location is no brake on $x$ for $f(x)=\sqrt[3]{x}$ considering you can accept the cube root of any real number. So the domain is all existent numbers, $(-\infty ,\infty ).$ When dealing with the set up of real numbers, you cannot take the square root of negative numbers. So $x$ -values are restricted for $f(x)=\sqrt[]{x}$ to nonnegative numbers and the domain is $[0,\infty ).$

5 .

Graph each formula of the piecewise function over its corresponding domain. Use the same calibration for the $\mathrm{ten}$ -axis and $y$ -axis for each graph. Bespeak inclusive endpoints with a solid circle and exclusive endpoints with an open up circle. Use an arrow to point $-\infty$ or $\infty .$ Combine the graphs to observe the graph of the piecewise function.

15 .

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

17 .

$(-\infty ,-\mathrm{xi})\cup (-11,2)\cup (\mathrm{ii},\infty )$

xix .

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

25 .

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

27 .

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

29 .

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

31 .

domain: $[-5,\mathrm{three}),$ range: $\left[0,2\right]$

33 .

domain: $(-\infty ,\mathrm{i}],$ range: $[0,\infty )$

35 .

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

37 .

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

39 .

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

41 .

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

43 .

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

45 .

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

47 .

$\begin{array}{cccc}f(-3)=1;& f(-2)=0;& f(-1)=0;& f(0)=0\end{array}$

49 .

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

51 .

$\begin{array}{cccc}f(-1)=-5;& f(0)=\mathrm{three};& f(2)=\mathrm{three};& f(4)=16\end{array}$

53 .

domain: $(-\infty ,1)\cup (\mathrm{ane},\infty )$

55 .

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

window: $[0.1,0.5];$ range: $[4,100]$

59 .

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

one.3 Section Exercises

i .

Yep, the average rate of modify of all linear functions is constant.

3 .

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

eleven .

$\frac{-1}{13\left(\mathrm{xiii}+h\right)}$

xiii .

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

19 .

increasing on $\left(-\infty ,-\mathrm{2.v}\right)\cup \left(1,\infty \right),$ decreasing on $(-2.5,1)$

21 .

increasing on $\left(-\infty ,\mathrm{one}\right)\cup \left(3,4\right),$ decreasing on $\left(1,\mathrm{three}\right)\cup \left(4,\infty \right)$

23 .

local maximum: $(-\mathrm{iii},\mathrm{fifty}),$ local minimum: $(3,-50)$

25 .

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

35 .

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

37 .

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

39 .

Local maximum at $(-\mathrm{0.v},\mathrm{vi}),$ local minima at $(-\mathrm{three.25},-47)$ and $(\mathrm{ii.1},-32),$ decreasing on $(-\infty ,-3.25)$ and $(-0.5,\mathrm{2.one}),$ increasing on $(-3.25,-0.5)$ and $(\mathrm{ii.one},\infty )$

45 .

2.seven gallons per minute

47 .

approximately –0.six milligrams per day

1.four Section Exercises

1 .

Observe the numbers that make the part in the denominator $g$ equal to nix, and cheque for any other domain restrictions on $f$ and $g,$ such equally an even-indexed root or zeros in the denominator.

three .

Yes. Sample answer: Allow $f(x)=x+\mathrm{ane}\text{ and}g(x)=\mathrm{ten}-1.$ Then $f(k(x))=f(\mathrm{10}-1)=(x-1)+1=x$ and $g(f(\mathrm{10}))=g(\mathrm{10}+\mathrm{ane})=(\mathrm{ten}+1)-1=\mathrm{ten}.$ So $f\circ g=\mathrm{grand}\circ f.$

5 .

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

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

$(fg)\left(x\right)=-{\mathrm{10}}^4-\mathrm{ii}{\mathrm{10}}^3+\mathrm{half-dozen}{x}^{\mathrm{two}}+12x,$ domain: $(-\infty ,\infty )$

$\left(\frac{f}{\mathrm{one\; thousand}}\right)\left(x\right)=\frac{x^2+\mathrm{two}\mathrm{10}}{6-x^2},$ domain: $(-\infty ,-\sqrt{6})\cup (-\sqrt{6},\sqrt{\mathrm{vi}})\cup (\sqrt{6},\infty )$

vii .

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

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

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

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

9 .

$(f+\mathrm{chiliad})(x)=\mathrm{iii}{x}^{\mathrm{two}}+\sqrt{x-5},$ domain: $[\mathrm{five},\infty )$

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

$(fg)(x)=3{\mathrm{ten}}^{\mathrm{ii}}\sqrt{x-5},$ domain: $[\mathrm{v},\infty )$

$\left(\frac{f}{g}\right)(x)=\frac{3x^2}{\sqrt{\mathrm{10}-5}},$ domain: $(\mathrm{five},\infty )$

eleven .

1. ⓐ 3
2. ⓑ $f\left(\mathrm{yard}\left(x\right)\right)=2{\left(3\mathrm{ten}-\mathrm{five}\right)}^{\mathrm{two}}+\mathrm{ane};$
3. ⓒ $\mathrm{thousand}\left(f)\left(x\right)\right)=6{\mathrm{10}}^2-2;$
4. ⓓ $\left(g\circ g\right)(x)=\mathrm{iii}(3\mathrm{ten}-5)-5=\mathrm{nine}x-20;$
5. ⓔ $\left(f\circ f\right)\left(-\mathrm{two}\right)=163$

xiii .

$f(g(x))=\sqrt{x^2+3}+2,g(f(\mathrm{ten}))=x+4\sqrt{x}+7$

fifteen .

$f(\mathrm{thousand}(\mathrm{ten}))=\sqrt[\mathrm{iii}]{\frac{x+1}{x^{\mathrm{three}}}}=\frac{\sqrt[\mathrm{three}]{\mathrm{ten}+\mathrm{ane}}}{x},g(f(\mathrm{ten}))=\frac{\sqrt[\mathrm{iii}]{\mathrm{10}}+1}{\mathrm{10}}$

17 .

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

19 .

$f(k(h(x)))={\left(\frac{1}{x+\mathrm{three}}\right)}^2+\mathrm{i}$

21 .

• ⓐ Text $(\mathrm{thou}\circ f)(x)=-\frac{\mathrm{iii}}{\sqrt{2-4\mathrm{10}}};$
• ⓑ $\left(-\infty ,\frac{1}{\mathrm{two}}\right)$

23 .

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

27 .

sample: $\begin{array}{l}f(\mathrm{ten})=x^{\mathrm{three}}\\ g(x)=x-\mathrm{five}\end{array}$

29 .

sample: $\begin{array}{l}f(x)=\frac{4}{x}\hfill \\ g(\mathrm{ten})={(x+\mathrm{two})}^2\hfill \end{array}$

31 .

sample: $\begin{array}{l}f(\mathrm{10})=\sqrt[3]{x}\\ m(x)=\frac{1}{2\mathrm{ten}-3}\end{array}$

33 .

sample: $\begin{array}{l}f(x)=\sqrt[4]{\mathrm{ten}}\\ g(x)=\frac{\mathrm{three}x-2}{x+\mathrm{v}}\end{array}$

35 .

sample: $f(\mathrm{ten})=\sqrt{x}$
$\mathrm{thousand}(\mathrm{ten})=2x+6$

37 .

sample: $f(x)=\sqrt[3]{x}$
$g(x)=(x-1)$

39 .

sample: $f(\mathrm{10})={\mathrm{ten}}^3$
$g(x)=\frac{\mathrm{ane}}{\mathrm{ten}-2}$

41 .

sample: $f(x)=\sqrt{x}$
$g(\mathrm{10})=\frac{2\mathrm{10}-\mathrm{one}}{3x+\mathrm{iv}}$

73 .

$f(g(0))=27,\mathrm{grand}\left(f(0)\right)=-94$

75 .

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

77 .

$18{\mathrm{ten}}^2+\mathrm{lx}x+51$

79 .

$g\circ \mathrm{one\; thousand}(x)=\mathrm{ix}x+20$

87 .

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

89 .

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

93 .

$A(t)=\pi {\left(25\sqrt{t+\mathrm{ii}}\right)}^2$ and $A(\mathrm{ii})=\pi {\left(25\sqrt{4}\right)}^2=2500\pi$ square inches

95 .

$A(\mathrm{five})=\pi {\left(2(5)+1\right)}^2=121\pi$ square units

97 .

• ⓐ $N(T(t))=23{(\mathrm{v}t+1.5)}^{\mathrm{two}}-56(5t+1.5)+1;$
• ⓑ 3.38 hours

ane.five Section Exercises

one .

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.

three .

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

5 .

For a function $f,$ substitute $(-x)$ for $(x)$ in $f(x).$ Simplify. If the resulting function is the same equally the original part, $f(-\mathrm{ten})=f(x),$ so the function is even. If the resulting office is the opposite of the original function, $f(-x)=-f(x),$ then the original function is odd. If the function is non the same or the opposite, then the function is neither odd nor fifty-fifty.

7 .

$g(x)=|\mathrm{10}-\mathrm{ane}|-3$

9 .

$g(x)=\frac{1}{{(x+4)}^{\mathrm{ii}}}+2$

11 .

The graph of $f(\mathrm{ten}+43)$ is a horizontal shift to the left 43 units of the graph of $f.$

13 .

The graph of $f(x-4)$ is a horizontal shift to the correct 4 units of the graph of $f.$

fifteen .

The graph of $f(\mathrm{10})+\mathrm{viii}$ is a vertical shift upward eight units of the graph of $f.$

17 .

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

xix .

The graph of $f(x+4)-\mathrm{one}$ is a horizontal shift to the left 4 units and a vertical shift downwards 1 unit of the graph of $f.$

21 .

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

23 .

decreasing on $[0,\infty )$

31 .

$k(x)=f(x-1),h(\mathrm{10})=f(x)+\mathrm{i}$

33 .

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

35 .

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

37 .

$f(\mathrm{ten})={(\mathrm{10}-\mathrm{ii})}^2$

39 .

$f(x)=|x+3|-\mathrm{ii}$

43 .

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

45 .

$f(x)=\sqrt{-\mathrm{10}}+1$

53 .

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

55 .

The graph of $g$ is a vertical stretch by a factor of four of the graph of $f.$

57 .

The graph of $g$ is a horizontal compression by a gene of $\frac{\mathrm{one}}{5}$ of the graph of $f.$

59 .

The graph of $g$ is a horizontal stretch past a gene of 3 of the graph of $f.$

61 .

The graph of $k$ is a horizontal reflection across the $y$ -axis and a vertical stretch by a cistron of 3 of the graph of $f.$

63 .

$g(\mathrm{10})=|-\mathrm{four}x|$

65 .

$g(\mathrm{ten})=\frac{\mathrm{i}}{\mathrm{three}{(x+\mathrm{ii})}^{\mathrm{ii}}}-\mathrm{three}$

67 .

$g(\mathrm{ten})=\frac{1}{\mathrm{ii}}{(\mathrm{ten}-\mathrm{five})}^2+\mathrm{i}$

69 .

The graph of the part $f(\mathrm{10})=x^2$ is shifted to the left 1 unit, stretched vertically by a factor of four, and shifted downwardly v units.

71 .

The graph of $f(x)=\left|x\right|$ is stretched vertically past a cistron of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, so shifted vertically 3 units up.

73 .

The graph of the function $f(\mathrm{ten})={\mathrm{10}}^3$ is compressed vertically by a factor of $\frac{1}{2}.$

75 .

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

77 .

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

i.six Section Exercises

one .

Isolate the absolute value term and so that the equation is of the form $\left|A\right|=B.$ Grade one equation by setting the expression inside the absolute 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.

iii .

The graph of the absolute value office does not cross the $x$ -axis, then the graph is either completely higher up or completely below the $x$ -axis.

five .

Starting time determine the purlieus points by finding the solution(southward) of the equation. Use the boundary points to form possible solution intervals. Choose a exam value in each interval to determine which values satisfy the inequality.

vii .

$\left|x+\mathrm{iv}\right|=\frac{\mathrm{i}}{2}$

9 .

$|f(x)-8|<0.03$

13 .

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

15 .

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

17 .

$\left\{\frac{11}{5},\frac{29}{5}\right\}$

xix .

$\left\{\frac{\mathrm{v}}{\mathrm{ii}},\frac{\mathrm{vii}}{2}\right\}$

23 .

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

25 .

$\left(0,-\mathrm{viii}\right);\left(-6,0\right),\left(4,0\right)$

27 .

$\left(0,-\mathrm{vii}\right);$ no $x$ -intercepts

29 .

$(-\infty ,-\mathrm{eight})\cup (12,\infty )$

33 .

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

35 .

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

53 .

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

55 .

$\mathrm{ten}\text{-}$ intercepts:

59 .

There is no solution for $a$ that volition keep the office from having a $y$ -intercept. The absolute value function ever crosses the $y$ -intercept when $x=0.$

61 .

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

63 .

$\left|\mathrm{10}-5.0\right|\le 0.01$

ane.7 Section Exercises

i .

Each output of a function must have exactly one output for the function to be ane-to-i. If whatsoever horizontal line crosses the graph of a function more than once, that means that $y$ -values repeat and the function is not ane-to-1. If no horizontal line crosses the graph of the role more than once, then no $y$ -values echo and the office is one-to-one.

3 .

Yep. For example, $f(x)=\frac{\mathrm{one}}{\mathrm{ten}}$ is its own changed.

5 .

Given a function $y=f(\mathrm{10}),$ solve for $x$ in terms of $y.$ Interchange the $x$ and $y.$ Solve the new equation for $y.$ The expression for $y$ is the changed, $y=f^{-\mathrm{ane}}(x).$

vii .

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

nine .

$f^{-1}(x)=2-x$

11 .

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

xiii .

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

xv .

domain of $f(x):[0,\infty );f^{-1}(x)=\sqrt{x+\mathrm{five}}$

16 .

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

17 .

$f(\mathrm{chiliad}(\mathrm{ten}))=\mathrm{10},\mathrm{1000}(f(\mathrm{10}))=x$

41 .

$x$	ane	4	7	12	16
$f^{-1}(x)$	3	half-dozen	ix	xiii	14

43 .

$f^{-1}(x)={(\mathrm{one}+\mathrm{10})}^{\mathrm{i}/3}$

45 .

$f^{-\mathrm{i}}(x)=\frac{5}{\mathrm{nine}}\left(x-32\right).$ Given the Fahrenheit temperature, $x,$ this formula allows you to summate 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 .

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

17 .

$x=-\mathrm{1.viii}$ or $\text{ or}\mathrm{ten}=1.8$

xix .

$\frac{-64+\mathrm{eighty}a-16a^{\mathrm{two}}}{-\mathrm{ane}+a}=-16a+64$

21 .

$\left(-\infty ,-2\right)\cup \left(-2,6\right)\cup \left(6,\infty \right)$

27 .

increasing $\left(\mathrm{two},\infty \right);$ decreasing $(-\infty ,2)$

29 .

increasing $\left(-\mathrm{three},\mathrm{one}\right);$ abiding $(-\infty ,-3)\cup \left(\mathrm{one},\infty \right)$

31 .

local minimum $\left(-2,-3\right);$ local maximum $\left(1,\mathrm{iii}\right)$

33 .

Absolute Maximum: x

35 .

$\left(f\circ \mathrm{thousand}\right)(x)=17-\mathrm{xviii}x;\left(g\circ f\right)(\mathrm{10})=-\mathrm{seven}-\mathrm{xviii}\mathrm{10}$

37 .

$\left(f\circ g\right)(x)=\sqrt{\frac{1}{\mathrm{ten}}+2};$ $\left(\mathrm{1000}\circ f\right)(\mathrm{ten})=\frac{1}{\sqrt{\mathrm{10}+2}}$

39 .

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

41 .

$\left(f\circ g\right)(\mathrm{ten})=\frac{1}{\sqrt{x}},\mathrm{10}>0$

43 .

sample: $k(x)=\frac{2\mathrm{ten}-\mathrm{one}}{\mathrm{iii}\mathrm{ten}+4};f(\mathrm{10})=\sqrt{\mathrm{10}}$

55 .

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

63 .

$f(\mathrm{10})=\frac{1}{2}\left|\mathrm{ten}+\mathrm{two}\right|+\mathrm{one}$

65 .

$f(x)=-\mathrm{iii}\left|\mathrm{10}-3\right|+3$

69 .

$x=-22,\mathrm{10}=14$

71 .

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

73 .

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

77 .

The office is one-to-one.

78 .

The role is non ane-to-one.

Do Test

1 .

The relation is a function.

five .

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

xix .

$x=-7$ and $\mathrm{ten}=\mathrm{x}$

21 .

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

23 .

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

25 .

$\left(1.1,-0.9\right)$

29 .

$f(x)=\{\begin{array}{c}\left|x\right|\text{if}x\le \mathrm{two}\\ 3\text{if}x>\mathrm{two}\end{array}$

35 .

$f^{-1}(x)=-\frac{\mathrm{ten}-11}{\mathrm{two}}$

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

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