2nd Six Weeks Midterm Assessment Review Glencoe Precalculus

Endeavour It

i.ane Functions and Office Note

one.

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

6.

$y=f\left(x\right)=\frac{\sqrt[3]{x}}{2}$

9.

1. ⓐ yes, because each bank business relationship has a single balance at any given time
2. ⓑ no, because several depository financial institution account numbers may have the same remainder
3. ⓒ no, considering the same output may correspond to more than ane input.

10.

1. ⓐ Aye, letter grade is a function of percentage form;
2. ⓑ No, it is non i-to-one. In that location are 100 dissimilar pct numbers nosotros could get but merely about 5 possible letter grades, and so in that location cannot be only one per centum number that corresponds to each letter course.

12.

No, because information technology does non laissez passer the horizontal line test.

1.2 Domain and Range

1.

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

iii.

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

4.

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

5.

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

6.

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

7.

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

1.3 Rates of Change and Beliefs of Graphs

1.

$\frac{\$\mathrm{ii.84}-\$2.31}{\mathrm{five}\text{ years}}=\frac{\$0.53}{5\text{ years}}=\$0.106$ per year.

4.

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

ane.4 Limerick of Functions

1.

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

No, the functions are non the same.

ii.

A gravitational force is nevertheless a strength, so $a\left(G(r)\right)$ makes sense every bit the dispatch of a planet at a distance r from the Sun (due to gravity), simply $G\left(a(F)\right)$ does not make sense.

3.

$f(\mathrm{grand}(\mathrm{one}))=f(3)=3$ and $g(f(4))=\mathrm{chiliad}(1)=3$

4.

$\mathrm{1000}(f(\mathrm{ii}))=m(5)=3$

6.

$\left[-\mathrm{four},0\right)\cup \left(0,\infty \right)$

7.

Possible respond:

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

i.5 Transformation of Functions

1.

$$b(t)=h(t)+10=-\mathrm{4.ix}{t}^2+30t+10$$

2.

The graphs of $f(\mathrm{10})$ and $g(x)$ are shown beneath. The transformation is a horizontal shift. The part is shifted to the left past 2 units.

4.

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

six.

1. ⓐ

   $\mathrm{thou}(x)=-f(x)$

   $\mathrm{10}$	-two	0	ii	4
   $g(x)$	$-5$	$-\mathrm{x}$	$-15$	$-20$

2. ⓑ

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

   $\mathrm{ten}$	-2	0	two	four
   $h(x)$	fifteen	10	5	unknown

seven.

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

9.

$x$	2	4	vi	8
$g(x)$	9	12	xv	0

11.

$\mathrm{thousand}(\mathrm{10})=f\left(\frac{\mathrm{i}}{3}x\right)$ so using the square root function we become $g(\mathrm{10})=\sqrt{\frac{1}{\mathrm{iii}}\mathrm{ten}}$

ane.6 Absolute Value Functions

2.

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

three.

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

5.

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

seven.

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

ane.7 Changed Functions

4.

The domain of function $f^{-\mathrm{i}}$ is $(-\infty \text{,}-2)$ and the range of part $f^{-1}$ is $(1,\infty ).$

5.

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

viii.

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

1.one 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 have the same first coordinate.

3.

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

v.

When a horizontal line intersects the graph of a office more than one time, that indicates that for that output in that location is more than than one input. A part is ane-to-one if each output corresponds to only one input.

27.

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

29.

$f(-3)=\sqrt{\mathrm{v}}+5;$
$f(\mathrm{two})=5;$
$f(-a)=\sqrt{2+a}+5;$
$-f(a)=-\sqrt{2-a}-5;$
$f(a+h)=\sqrt{2-a-h}+5$

31.

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

33.

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

35.

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

37.

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

39.

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

53.

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

55.

non a part and so information technology is also not a one-to-one function

59.

function, merely not one-to-ane

67.

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

69.

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

71.

$\begin{array}{ccccc}f(-2)=\mathrm{iv};& f(-1)=4.414;& f(0)=4.732;& f(1)=5;& f(\mathrm{two})=5.236\end{array}$

73.

$\begin{array}{ccccc}f(-2)=\frac{\mathrm{one}}{\mathrm{ix}};& f(-\mathrm{i})=\frac{1}{3};& f(0)=\mathrm{i};& f(\mathrm{i})=3;& f(\mathrm{ii})=\mathrm{nine}\end{array}$

77.

$[0,\text{ 100}]$

79.

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

81.

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

83.

$[0,\text{ 10}]$

85.

$[\mathrm{-0.1},\text{0.ane}]$

87.

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

89.

1. ⓐ $\mathrm{chiliad}(5000)=\mathrm{l};$
2. ⓑ The number of cubic yards of dirt required for a garden of 100 square feet is 1.

91.

1. ⓐ The superlative of a rocket above basis after 1 second is 200 ft.
2. ⓑ the acme of a rocket higher up footing after two seconds is 350 ft.

ane.2 Section Exercises

one.

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

3.

There is no restriction on $\mathrm{ten}$ for $f(x)=\sqrt[3]{\mathrm{ten}}$ considering you tin take the cube root of any real number. So the domain is all real numbers, $(-\infty ,\infty ).$ When dealing with the gear up of real numbers, you cannot accept the square root of negative numbers. And then $x$ -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 respective domain. Utilise the aforementioned calibration for the $x$ -centrality and $y$ -axis for each graph. Point 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 find the graph of the piecewise role.

15.

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

17.

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

19.

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

25.

$\left(-\infty ,-9\right)\cup \left(-\mathrm{ix},9\right)\cup \left(9,\infty \right)$

27.

domain: $(\mathrm{two},8],$ range $[\mathrm{half-dozen},\mathrm{viii})$

29.

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

31.

domain: $[-5,3),$ range: $\left[0,2\right]$

33.

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

35.

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

37.

domain: $[-\mathrm{three},\infty );$ range: $[0,\infty )$

39.

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

41.

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

43.

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

45.

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

47.

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

49.

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

51.

$\begin{array}{cccc}f(-\mathrm{ane})=-\mathrm{v};& f(0)=3;& f(\mathrm{ii})=3;& f(4)=16\end{array}$

53.

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

55.

window: $[-0.5,-0.1];$ range: $[\mathrm{four},100]$

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

59.

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

1.3 Section Exercises

1.

Yep, the boilerplate rate of alter of all linear functions is constant.

iii.

The accented maximum and minimum chronicle to the entire graph, whereas the local extrema relate only to a specific region effectually an open interval.

11.

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

thirteen.

$\mathrm{three}{h}^2+9h+9$

xix.

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

21.

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

23.

local maximum: $(-3,50),$ local minimum: $(3,-50)$

25.

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

35.

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

37.

Local minimum at $(-2,-2),$ decreasing on $(-3,-2),$ increasing on $(-2,\infty )$

39.

Local maximum at $(-0.5,6),$ local minima at $(-\mathrm{iii.25},-47)$ and $(\mathrm{2.one},-32),$ decreasing on $(-\infty ,-3.25)$ and $(-0.5,\mathrm{two.1}),$ increasing on $(-3.25,-\mathrm{0.five})$ and $(2.1,\infty )$

45.

two.7 gallons per infinitesimal

47.

approximately –0.vi milligrams per day

1.four Section Exercises

1.

Find the numbers that brand the part in the denominator $g$ equal to zero, and check for whatever other domain restrictions on $f$ and $g,$ such as an even-indexed root or zeros in the denominator.

3.

Yeah. Sample answer: Let $f(x)=\mathrm{ten}+1\text{ and}\mathrm{grand}(x)=\mathrm{10}-\mathrm{one.}$ Then $f(g(\mathrm{ten}))=f(\mathrm{ten}-1)=(x-1)+1=\mathrm{ten}$ and $\mathrm{chiliad}(f(x))=g(x+1)=(\mathrm{10}+1)-\mathrm{i}=\mathrm{10}.$ So $f\circ g=g\circ f.$

5.

$(f+g)\left(x\right)=2x+6,$ domain: $(-\infty ,\infty )$

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

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

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

7.

$(f+\mathrm{thou})\left(\mathrm{10}\right)=\frac{4x^3+8{\mathrm{ten}}^2+1}{2\mathrm{ten}},$ domain: $(-\infty ,0)\cup (0,\infty )$

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

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

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

9.

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

$(f-\mathrm{yard})(\mathrm{ten})=\mathrm{iii}{x}^2-\sqrt{\mathrm{10}-\mathrm{v}},$ domain: $[5,\infty )$

$(f\mathrm{chiliad})(\mathrm{10})=\mathrm{three}{x}^{\mathrm{ii}}\sqrt{x-5},$ domain: $[5,\infty )$

$\left(\frac{f}{g}\right)(\mathrm{ten})=\frac{\mathrm{three}{x}^{\mathrm{two}}}{\sqrt{x-5}},$ domain: $(\mathrm{v},\infty )$

11.

1. ⓐ iii
2. ⓑ $f\left(g\left(x\right)\right)=2{\left(3\mathrm{ten}-5\right)}^2+\mathrm{ane};$
3. ⓒ $g\left(f)\left(x\right)\right)=\mathrm{six}{x}^{\mathrm{two}}-2;$
4. ⓓ $\left(\mathrm{thou}\circ \mathrm{one\; thousand}\right)(\mathrm{10})=\mathrm{three}(3x-\mathrm{five})-5=9\mathrm{ten}-20;$
5. ⓔ $\left(f\circ f\right)\left(-2\right)=163$

13.

$f(g(x))=\sqrt{x^2+3}+\mathrm{ii},\mathrm{1000}(f(\mathrm{ten}))=\mathrm{10}+4\sqrt{\mathrm{ten}}+\mathrm{vii}$

fifteen.

$f(g(x))=\sqrt[\mathrm{iii}]{\frac{x+\mathrm{ane}}{x^{\mathrm{three}}}}=\frac{\sqrt[3]{x+\mathrm{i}}}{\mathrm{ten}},g(f(x))=\frac{\sqrt[\mathrm{iii}]{\mathrm{ten}}+\mathrm{ane}}{x}$

17.

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

19.

$f(g(h(\mathrm{10})))={\left(\frac{\mathrm{ane}}{x+3}\right)}^2+\mathrm{i}$

21.

• ⓐ Text $(k\circ f)(x)=-\frac{3}{\sqrt{\mathrm{ii}-\mathrm{iv}\mathrm{10}}};$
• ⓑ $\left(-\infty ,\frac{1}{2}\right)$

23.

1. ⓐ $(0,\mathrm{ii})\cup (2,\infty );$
2. ⓑ $(-\infty ,-\mathrm{two})\cup (2,\infty );$ c. $(0,\infty )$

27.

sample: $\begin{array}{l}f(x)={\mathrm{ten}}^3\\ g(x)=\mathrm{10}-\mathrm{five}\end{array}$

29.

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

31.

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

33.

sample: $\begin{array}{l}f(\mathrm{ten})=\sqrt[\mathrm{four}]{\mathrm{10}}\\ \mathrm{grand}(\mathrm{10})=\frac{3\mathrm{ten}-2}{x+5}\end{array}$

35.

sample: $f(\mathrm{10})=\sqrt{x}$
$g(\mathrm{ten})=2\mathrm{ten}+\mathrm{six}$

37.

sample: $f(x)=\sqrt[3]{x}$
$\mathrm{1000}(x)=(x-1)$

39.

sample: $f(x)={\mathrm{ten}}^3$
$g(\mathrm{ten})=\frac{1}{x-2}$

41.

sample: $f(x)=\sqrt{x}$
$g(x)=\frac{2x-1}{\mathrm{three}x+\mathrm{four}}$

73.

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

75.

$f(\mathrm{one\; thousand}(0))=\frac{\mathrm{i}}{5},g(f(0))=5$

77.

$\mathrm{eighteen}{\mathrm{ten}}^2+\mathrm{sixty}x+51$

79.

$\mathrm{thou}\circ g(x)=\mathrm{nine}x+\mathrm{xx}$

87.

$(f\circ g)(6)=\mathrm{half-dozen}$ ; $(\mathrm{thousand}\circ f)(\mathrm{half-dozen})=6$

89.

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

93.

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

95.

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

97.

• ⓐ $N(T(t))=23{(\mathrm{v}t+\mathrm{i.five})}^2-56(\mathrm{v}t+1.5)+1;$
• ⓑ 3.38 hours

1.v Section Exercises

ane.

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.

3.

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

five.

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

7.

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

nine.

$g(x)=\frac{1}{{(\mathrm{10}+4)}^{\mathrm{two}}}+2$

eleven.

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

13.

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

15.

The graph of $f(x)+8$ is a vertical shift upwardly 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 $f.$

nineteen.

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

21.

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

23.

decreasing on $[0,\infty )$

31.

$\mathrm{one\; thousand}(x)=f(\mathrm{10}-1),h(\mathrm{10})=f(x)+1$

33.

$f(\mathrm{ten})=|\mathrm{10}-\mathrm{three}|-\mathrm{ii}$

35.

$f(x)=\sqrt{x+\mathrm{iii}}-\mathrm{ane}$

37.

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

39.

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

43.

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

45.

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

53.

The graph of $\mathrm{thou}$ 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 compression by a factor of $\frac{\mathrm{i}}{5}$ of the graph of $f.$

59.

The graph of $\mathrm{chiliad}$ is a horizontal stretch by a cistron of 3 of the graph of $f.$

61.

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

63.

$g(x)=|-4x|$

65.

$g(x)=\frac{1}{3{(x+2)}^2}-3$

67.

$g(\mathrm{ten})=\frac{1}{2}{(\mathrm{10}-5)}^2+1$

69.

The graph of the function $f(x)=x^2$ is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

71.

The graph of $f(\mathrm{ten})=\left|x\right|$ is stretched vertically past a factor of two, shifted horizontally 4 units to the right, reflected across the horizontal axis, and and then shifted vertically 3 units up.

73.

The graph of the function $f(\mathrm{ten})=x^{\mathrm{three}}$ is compressed vertically by a cistron of $\frac{\mathrm{i}}{2}.$

75.

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

77.

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

1.half-dozen Section Exercises

1.

Isolate the absolute value term and so that the equation is of the form $\left|A\right|=B.$ Class ane equation past setting the expression inside the absolute value symbol, $A,$ equal to the expression on the other side of the equation, $B.$ Grade 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 non cross the $x$ -axis, and then the graph is either completely in a higher place or completely beneath the $x$ -axis.

5.

First decide the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a examination value in each interval to determine which values satisfy the inequality.

7.

$\left|x+4\right|=\frac{1}{\mathrm{ii}}$

ix.

$|f(\mathrm{ten})-8|<0.03$

13.

$\{-\frac{9}{4},\frac{\mathrm{thirteen}}{4}\}$

15.

$\left\{\frac{\mathrm{ten}}{\mathrm{three}},\frac{\mathrm{xx}}{\mathrm{iii}}\right\}$

17.

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

19.

$\left\{\frac{5}{\mathrm{two}},\frac{7}{\mathrm{two}}\right\}$

23.

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

25.

$\left(0,-8\right);\left(-\mathrm{half-dozen},0\right),\left(4,0\right)$

27.

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

29.

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

33.

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

35.

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

53.

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

55.

$\mathrm{10}\text{-}$ 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.

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

i.7 Section Exercises

1.

Each output of a part must accept exactly one output for the function to be 1-to-one. If any horizontal line crosses the graph of a function more than once, that ways that $y$ -values repeat and the function is non one-to-1. If no horizontal line crosses the graph of the role more than once, then no $y$ -values repeat and the function is one-to-one.

3.

Yes. For example, $f(x)=\frac{1}{\mathrm{10}}$ is its own inverse.

v.

Given a role $y=f(x),$ 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, $y=f^{-1}(\mathrm{ten}).$

seven.

$f^{-\mathrm{one}}(\mathrm{ten})=\mathrm{10}-3$

9.

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

xi.

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

thirteen.

domain of $f(\mathrm{ten}):[-7,\infty );f^{-1}(x)=\sqrt{x}-7$

15.

domain of $f(\mathrm{10}):[0,\infty );f^{-\mathrm{i}}(x)=\sqrt{x+\mathrm{five}}$

16.

• ⓐ $f(\mathrm{1000}(x))=x$ and $g(f(x))=x.$
• ⓑ This tells usa that $f$ and $g$ are inverse functions

17.

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

41.

$x$	1	4	7	12	16
$f^{-1}(\mathrm{10})$	iii	half-dozen	9	13	fourteen

43.

$f^{-\mathrm{one}}(\mathrm{ten})={(1+\mathrm{ten})}^{1/\mathrm{iii}}$

45.

$f^{-\mathrm{one}}(\mathrm{10})=\frac{5}{\mathrm{nine}}\left(x-32\right).$ Given the Fahrenheit temperature, $x,$ this formula allows you to calculate the Celsius temperature.

47.

$t(d)=\frac{d}{50},$ $t(180)=\frac{180}{50}.$ The time for the car to travel 180 miles is three.6 hours.

Review Exercises

five.

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

17.

$\mathrm{10}=-\mathrm{1.viii}$ or $\text{ or}x=1.8$

19.

$\frac{-64+80a-\mathrm{sixteen}{a}^{\mathrm{two}}}{-1+a}=-\mathrm{xvi}a+64$

21.

$\left(-\infty ,-2\right)\cup \left(-2,6\right)\cup \left(\mathrm{half\; dozen},\infty \right)$

27.

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

29.

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

31.

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

35.

$\left(f\circ \mathrm{yard}\right)(\mathrm{10})=17-\mathrm{eighteen}x;\left(\mathrm{yard}\circ f\right)(x)=-7-\mathrm{eighteen}x$

37.

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

39.

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

41.

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

43.

sample: $\mathrm{grand}(x)=\frac{2x-1}{3\mathrm{ten}+4};f(\mathrm{10})=\sqrt{x}$

55.

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

63.

$f(x)=\frac{1}{2}\left|\mathrm{10}+2\right|+\mathrm{ane}$

65.

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

69.

$x=-22,x=14$

71.

$\left(-\frac{\mathrm{v}}{\mathrm{iii}},\mathrm{three}\right)$

73.

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

77.

The role is one-to-ane.

78.

The function is non ane-to-1.

Practice Test

1.

The relation is a role.

5.

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

19.

$x=-7$ and $x=10$

21.

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

23.

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

25.

$\left(\mathrm{1.one},-0.9\right)$

29.

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

35.

$f^{-\mathrm{ane}}(\mathrm{10})=-\frac{\mathrm{ten}-11}{2}$

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