In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.

$$

f(x)=\frac{2-x}{x-5}

$$

a) $\{x | x \neq-5, x \neq 2\}$

b) $\{x | x \neq 3\}$

c) $\{x | x \neq-2\}$

d) $\{x | x \neq-3\}$

e) $\{x | x \neq 5\}$

f) $\{x | x \neq-2, x \neq 5\}$

g) $\{x | x \neq 2\}$

h) $\{x | x \neq-2, x \neq-5\}$

i) $\quad\{x | x \neq 2, x \neq 5\}$

j) $\{x | x \neq-5\}$

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In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.

$$

f(x)=\frac{x+5}{x+2}

$$

a) $\{x | x \neq-5, x \neq 2\}$

b) $\{x | x \neq 3\}$

c) $\{x | x \neq-2\}$

d) $\{x | x \neq-3\}$

e) $\{x | x \neq 5\}$

f) $\{x | x \neq-2, x \neq 5\}$

g) $\{x | x \neq 2\}$

h) $\{x | x \neq-2, x \neq-5\}$

i) $\quad\{x | x \neq 2, x \neq 5\}$

j) $\{x | x \neq-5\}$

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In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.

$$

g(x)=\frac{x-3}{(x-2)(x-5)}

$$

a) $\{x | x \neq-5, x \neq 2\}$

b) $\{x | x \neq 3\}$

c) $\{x | x \neq-2\}$

d) $\{x | x \neq-3\}$

e) $\{x | x \neq 5\}$

f) $\{x | x \neq-2, x \neq 5\}$

g) $\{x | x \neq 2\}$

h) $\{x | x \neq-2, x \neq-5\}$

i) $\quad\{x | x \neq 2, x \neq 5\}$

j) $\{x | x \neq-5\}$

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In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.

$$

g(x)=\frac{x+3}{(x+2)(x-5)}

$$

a) $\{x | x \neq-5, x \neq 2\}$

b) $\{x | x \neq 3\}$

c) $\{x | x \neq-2\}$

d) $\{x | x \neq-3\}$

e) $\{x | x \neq 5\}$

f) $\{x | x \neq-2, x \neq 5\}$

g) $\{x | x \neq 2\}$

h) $\{x | x \neq-2, x \neq-5\}$

i) $\quad\{x | x \neq 2, x \neq 5\}$

j) $\{x | x \neq-5\}$

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In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.

$$

-h(x)=\frac{(x-2)(x-3)}{x+3}

$$

a) $\{x | x \neq-5, x \neq 2\}$

b) $\{x | x \neq 3\}$

c) $\{x | x \neq-2\}$

d) $\{x | x \neq-3\}$

e) $\{x | x \neq 5\}$

f) $\{x | x \neq-2, x \neq 5\}$

g) $\{x | x \neq 2\}$

h) $\{x | x \neq-2, x \neq-5\}$

i) $\quad\{x | x \neq 2, x \neq 5\}$

j) $\{x | x \neq-5\}$

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In each of Exercises 1–6, match the function described with the appropriate domain from the column on the right. Some choices of domain will not be used.

$$

f(x)=\frac{(x+2)(x+3)}{x-3}

$$

a) $\{x | x \neq-5, x \neq 2\}$

b) $\{x | x \neq 3\}$

c) $\{x | x \neq-2\}$

d) $\{x | x \neq-3\}$

e) $\{x | x \neq 5\}$

f) $\{x | x \neq-2, x \neq 5\}$

g) $\{x | x \neq 2\}$

h) $\{x | x \neq-2, x \neq-5\}$

i) $\quad\{x | x \neq 2, x \neq 5\}$

j) $\{x | x \neq-5\}$

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Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by

$$

H(t)=\frac{t^{2}+3 t}{2 t+3}

$$

can be used to determine how long it would take if they worked together.

How long will it take them, working together, to complete a day's orders if Molly can process the orders alone in 5 hr?

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Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by

$$

H(t)=\frac{t^{2}+3 t}{2 t+3}

$$

can be used to determine how long it would take if they worked together.

How long will it take them, working together, to complete a day's orders if Molly can process the srders alone in 7 hr?

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Jasmine usually takes 3 hr more than Molly does to process a day’s orders at Books To Go. If Molly takes t hr to process a day’s orders, the function given by

$$

H(t)=\frac{t^{2}+3 t}{2 t+3}

$$

can be used to determine how long it would take if they worked together.

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For each rational function, find the function values indicated, provided the value exists.

$$

v(t)=\frac{4 t^{2}-5 t+2}{t+3} ; v(0), v(-2), v(7)

$$

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For each rational function, find the function values indicated, provided the value exists.

$$

g(x)=\frac{2 x^{3}-9}{x^{2}-4 x+4} ; g(0), g(2), g(-1)

$$

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For each rational function, find the function values indicated, provided the value exists.

$$

r(t)=\frac{t^{2}-5 t+4}{t^{2}-9} ; r(1), r(2), r(-3)

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{25}{-7 x}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{14}{-5 y}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{t-3}{t+8}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{a-8}{a+7}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{a}{3 a-12}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{x^{2}}{4 x-12}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{x^{2}-16}{x^{2}-3 x-28}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{p^{2}-9}{p^{2}-7 p+10}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{m^{3}-2 m}{m^{2}-25}

$$

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List all numbers for which each rational expression is undefined.

$$

\frac{7-3 x+x^{2}}{49-x^{2}}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{18 t^{3} w^{2}}{27 t^{7} w}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{8 y^{5} z}{4 y^{9} z^{3}}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{3 x^{2}-12 x}{3 x^{2}+15 x}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{4 y^{2}-20 y}{4 y^{2}+12 y}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{6 a^{2}-3 a}{7 a^{2}-7 a}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{3 m^{2}+3 m}{6 m^{2}+9 m}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{3 a^{2}+9 a-12}{6 a^{2}-30 a+24}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{2 t^{2}-6 t+4}{4 t^{2}+12 t-16}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{x^{2}+8 x+16}{x^{2}-16}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{x^{2}-25}{x^{2}-10 x+25}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{5 x^{2}-20}{10 x^{2}-40}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{6 x^{2}-54}{4 x^{2}-36}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{a^{2}-25}{a^{2}+10 a+25}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{a^{2}-16}{a^{2}-8 a+16}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{7 s^{2}-28 t^{2}}{28 t^{2}-7 s^{2}}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{9 m^{2}-4 n^{2}}{4 n^{2}-9 m^{2}}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{3 y^{3}+24}{y^{2}-2 y+4}

$$

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Simplify by removing a factor equal to 1.

$$

\frac{x^{3}-27}{5 x^{2}+15 x+45}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

f(x)=\frac{3 x+21}{x^{2}+7 x}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

f(x)=\frac{5 x+20}{x^{2}+4 x}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

g(x)=\frac{x^{2}-9}{5 x+15}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

g(x)=\frac{8 x-16}{x^{2}-4}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

h(x)=\frac{4-x}{5 x-20}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

h(x)=\frac{7-x}{3 x-21}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

f(t)=\frac{t^{2}-9}{t^{2}+4 t+3}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

f(t)=\frac{t^{2}-25}{t^{2}-6 t+5}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

g(t)=\frac{21-7 t}{3 t-9}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

g(t)=\frac{12-6 t}{5 t-10}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

h(t)=\frac{t^{2}+5 t+4}{t^{2}-8 t-9}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

h(t)=\frac{t^{2}-3 t-4}{t^{2}+9 t+8}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

f(x)=\frac{9 x^{2}-4}{3 x-2}

$$

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Write simplified form for each of the following. Be sure to list all restrictions on the domain, as in Example 7.

$$

f(x)=\frac{4 x^{2}-1}{2 x-1}

$$

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Determine the vertical asymptotes of the graph of each function.

$$

f(x)=\frac{3 x-12}{3 x+15}

$$

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Determine the vertical asymptotes of the graph of each function.

$$

f(x)=\frac{4 x-20}{4 x+12}

$$

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Determine the vertical asymptotes of the graph of each function.

$$

g(x)=\frac{12-6 x}{5 x-10}

$$

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Determine the vertical asymptotes of the graph of each function.

$$

r(x)=\frac{21-7 x}{3 x-9}

$$

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Determine the vertical asymptotes of the graph of each function.

$$

t(x)=\frac{x^{3}+3 x^{2}}{x^{2}+6 x+9}

$$

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Determine the vertical asymptotes of the graph of each function.

$$

g(x)=\frac{x^{2}-4}{2 x^{2}-5 x+2}

$$

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Determine the vertical asymptotes of the graph of each function.

$$

f(x)=\frac{x^{2}-x-6}{x^{2}-6 x+8}

$$

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Determine the vertical asymptotes of the graph of each function.

$$

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

$$

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In Exercises 79–84, match each function with one of the following graphs.

$$

h(x)=\frac{1}{x}

$$

(GRAPH CANT COPY)

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In Exercises 79–84, match each function with one of the following graphs.

$$

q(x)=-\frac{1}{x}

$$

(GRAPH CANT COPY)

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In Exercises 79–84, match each function with one of the following graphs.

In Exercises 79–84, match each function with one of the following graphs.

$$

h(x)=\frac{1}{x}

$$

(GRAPH CANT COPY)

(GRAPH CANT COPY)

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In Exercises 79–84, match each function with one of the following graphs.

$$

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

$$

(GRAPH CANT COPY)

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In Exercises 79–84, match each function with one of the following graphs.

$$

r(x)=\frac{4 x-2}{x^{2}-2 x+1}

$$

(GRAPH CANT COPY)

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Explain why the graphs of $f(x)=5 x$ and $g(x)=\frac{5 x^{2}}{x}$ differ

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Explain why the graphs of $f(x)=5 x$ and $g(x)=\frac{5 x^{2}}{x}$ differ.

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If a rational expression is undefined for $x=5$ and $x=-3,$ what is the degree of the denominator? Why?

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To prepare for Section 7.2, review multiplication and division using fraction notation.

Simplify.

$$

-\frac{2}{15} \cdot \frac{10}{7}[1.7]

$$

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To prepare for Section 7.2, review multiplication and division using fraction notation.

Simplify.

$$

\left(\frac{3}{4}\right)\left(\frac{-20}{9}\right)[1.7]

$$

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To prepare for Section 7.2, review multiplication and division using fraction notation.

Simplify.

$$

\frac{7}{10} \div\left(-\frac{8}{15}\right)[1.7]

$$

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Simplify.

$$

\frac{7}{10} \div\left(-\frac{8}{15}\right)[1.7]

$$

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To prepare for Section 7.2, review multiplication and division using fraction notation.

Simplify.

$$

\frac{7}{9}-\frac{2}{3} \cdot \frac{6}{7}[1.8]

$$

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To prepare for Section 7.2, review multiplication and division using fraction notation.

Simplify.

$$

\frac{2}{3}-\left(\frac{3}{4}\right)^{2}

$$

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Keith incorrectly simplifies

$$

\frac{x^{2}+x-2}{x^{2}+3 x+2} \text { as } \frac{x-1}{x+2}

$$

He then checks his simplification by evaluating both expressions for $x=1 .$ Use this situation to explain why evaluating is not a foolproof check.

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How could you convince someone that $a-b$ and $b-a$ are opposites of each other?

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Calculate the slope of the line passing through $(a, f(a))$ and $(a+h, f(a+h))$ for the function $f$ given by $f(x)=x^{2}+5 .$ Be sure your answer is simplified.

(GRAPH CANT COPY)

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Calculate the slope of the line passing through the points $(a, f(a))$ and $(a+h, f(a+h))$ for the function $f$ given by $f(x)=3 x^{2} .$ Be sure your answer is simplified.

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Simplify.

$$

\frac{(x-1)\left(x^{4}-1\right)\left(x^{2}-1\right)}{\left(x^{2}+1\right)(x-1)^{2}\left(x^{4}-2 x^{2}+1\right)}

$$

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Simplify.

$$

\frac{\left(t^{4}-1\right)\left(t^{2}-9\right)(t-9)^{2}}{\left(t^{4}-81\right)\left(t^{2}+1\right)(t+1)^{2}}

$$

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Simplify.

$$

\frac{(t+2)^{3}\left(t^{2}+2 t+1\right)(t+1)}{(t+1)^{3}\left(t^{2}+4 t+4\right)(t+2)}

$$

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Simplify.

$$

\frac{\left(x^{2}-y^{2}\right)\left(x^{2}-2 x y+y^{2}\right)}{(x+y)^{2}\left(x^{2}-4 x y-5 y^{2}\right)}

$$

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Determine the domain and the range of each function from its graph.

(GRAPH CANT COPY)

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Determine the domain and the range of each function from its graph.

(GRAPH CANT COPY)

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Determine the domain and the range of each function from its graph.

(GRAPH CANT COPY)

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Graph the function given by

$$

f(x)=\frac{x^{2}-9}{x-3}

$$

(Hint. Determine the domain of $f$ and simplify.)

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Select any number $x$, multiply by $2,$ add $5,$ multiply by $5,$ subtract $25,$ and divide by $10 .$ What do you get? Explain how this procedure can be used for a number trick.

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