Piecewise Functions

1.


A	B
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C	D

Choose the graph that represents the piecewise function

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$f(x) = \left\{ \begin{array}{lcl} \frac{1}{2}x - 2 \amp \quad \amp \mbox{for } x \leq 1 \\ -\frac{3}{4}x - 4 \amp \quad \amp \mbox{for } x > 1 \\ \end{array}\right.$

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from the above choices. You may click a graph to enlarge it.

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Correct Letter:

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Answer.

$\text{A}$

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2.

Let $y = f(x)$ be the piecewise defined function given below.

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\begin{equation*} f(x) = \left\lbrace \begin{array}{lcl} 1-x, \amp \amp \mbox{ if } x \leq -1, \\ 2, \amp \amp \mbox{ if } -1 \lt x \lt 1, \\ x+1, \amp \amp \mbox{ if } x \geq 1. \end{array} \right. \end{equation*}

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a. $f(-2) =$

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b. $f(1) =$

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c. For what values of $x$ is $f(x) = 2\text{?}$

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d. Find the domain and range of $f\text{.}$ (You may find it helpful to graph this function on your own paper to find the domain and range.) Your answers must be inequalities (not intervals).

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Domain:

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Range:

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Answer 1.

$3$

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Answer 2.

$2$

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Answer 3.

$-1\le x\le 1$

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Answer 4.

$-\infty < x < \infty $

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Answer 5.

$y\ge 2$

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3.

A restaurant offers a catering service which costs $21.00 per person with a $107.75 service charge. For parties of 55 or more people, a group discount applies, and the cost is $17.75 per person along with the service charge dropping to $53.00. Write a piecewise-defined linear function which calculates the total cost $T$ of the catering service which serves $n$ people.

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$\displaystyle T(n) = \left{ \begin{array}{cc} \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \end{array}\right.$

if $n$ $\geq$

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Answer 1.

$21n+107.75$

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Answer 2.

$1$

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Answer 3.

$55-1$

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Answer 4.

$17.75n+53$

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Answer 5.

$55$

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4.

The graph of a piecewise function, $f(x)\text{,}$ is depicted above. Find its equation:

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$\displaystyle f(x) = \left{ \begin{array}{cc} \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \end{array}\right.$

for $x$

select

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<=

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<

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for

select

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<=

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<

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$x$

select

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<=

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<

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for $x$

select

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>=

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>
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Remarks: Note that “<=” stands for “less than or equal” and “>=” stands for “greater than or equal.” You must have your browser window expanded wide enough to correctly display the above formula.

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Answer 1.

$-3$

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Answer 2.

${\verb!<=!}$

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Answer 3.

$-4$

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Answer 4.

$1.5\mathopen{}\left(x+4\right)-1$

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Answer 5.

$-4$

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Answer 6.

${\verb!<!}$

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Answer 7.

${\verb!<=!}$

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Answer 8.

$2$

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Answer 9.

$1.11111\mathopen{}\left(x-2\right)-11$

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Answer 10.

${\verb!>!}$

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Answer 11.

$2$

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5.

Find the following values of the function

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\begin{equation*} f(x) = \left\{ \begin{array}{ccc} x + 8 \amp \amp x \leq 2 \\ 3 - x \amp \amp x > 2 \end{array} \right. \end{equation*}

$f(-2) =$

$f(2) =$

$f(6) =$

Answer 1.

$6$

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Answer 2.

$10$

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Answer 3.

$-3$

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6.

Given the graph of $f(x)$ above, find the following:

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(a) Domain:

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(b) Range:

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Write your answer using interval notation. Help:

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Answer 1.

$\left(-\infty ,0\right)\cup \left[4,7\right]$

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Answer 2.

$\left(-\infty ,3\right]$

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7.

Given the graph of $f(x)$ above, find the following and write your answers using interval notation:

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(a) Domain:

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(b) Range:

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(c) Interval(s) on which $f(x)$ is increasing:

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(d) Interval(s) on which $f(x)$ is decreasing:

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(e) Interval(s) on which $f(x)$ is constant:

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(f) Relative maxima:

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(g) Relative minima:

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Answer 1.

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

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Answer 2.

$\left[-3,\infty \right)$

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Answer 3.

$\left(-7,-4\right)$

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Answer 4.

$\left(-\infty ,-7\right), \left(0,2\right)$

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Answer 5.

$\left(-4,0\right), \left(2,6\right)$

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Answer 6.

$\text{none}$

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Answer 7.

$-3$

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8.


A	B
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C	D

Choose the graph that represents the piecewise function

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\begin{equation*} f(x) = \left{\begin{array}{ccl} -2 \amp \quad \amp \mbox{for } x \leq -7 \\ -\frac{3}{2}x - 1 \amp \quad \amp \mbox{for } -7 \lt x \leq -1 \\ x - 5 \amp \quad \amp \mbox{for } x > -1 \\ \end{array} \right. \end{equation*}

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from the above choices. You may click a graph to enlarge it.

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Correct Letter:

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Answer.

$\text{C}$

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9.

The graph of a piecewise function, $f(x)\text{,}$ is depicted above. Find its equation:

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$\displaystyle f(x) = \left{ \begin{array}{cc} \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \amp \\ \end{array}\right.$

for $x$

select

🔗
<=

🔗
<

🔗

for

select

🔗
<=

🔗
<

🔗

$x$

select

🔗
<=

🔗
<

🔗

for $x$

select

🔗
>=

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>
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Answer 1.

$11$

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Answer 2.

${\verb!<!}$

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Answer 3.

$-8$

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Answer 4.

$-\left(0.181818\mathopen{}\left(x+8\right)+2\right)$

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Answer 5.

$-8$

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Answer 6.

${\verb!<=!}$

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Answer 7.

${\verb!<!}$

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Answer 8.

$3$

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Answer 9.

$-\left(0.777778\mathopen{}\left(x-3\right)+1\right)$

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Answer 10.

${\verb!>=!}$

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Answer 11.

$3$

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10.

Find $\small{f(0), f(7), f(-7), f(3),}$ and $\small{f(\sqrt{7}),}$ for:

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\begin{equation*} \small{f(x) = \begin{cases}\displaystyle{\frac{2}{x}}\amp \text{if}\ x > 3\cr \displaystyle{2x}\amp \text{if}\ x \le 3\end{cases}} \end{equation*}

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$\small{f(0)}$	$=$
$\small{f(7)}$	$=$
$\small{f(-7)}$	$=$
$\small{f(3)}$	$=$
$\small{f(\sqrt{7})}$	$=$

Answer 1.

$0$

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Answer 2.

$0.285714$

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Answer 3.

$-14$

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Answer 4.

$6$

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Answer 5.

$5.2915$

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Solution.

$\small{x}\text{,}$$\small{x}$$\small{x}\text{.}$$\small{f(7), 7 > 3}$$\small{f(7) = \large{\frac{2}{7}}}\text{.}$$\small{\le 3}\text{,}$

\begin{align*} \small{f(0)} \amp = \small{2 (0) = 0}\\ \small{f(-7)} \amp = \small{2 (-7) = -14} \\ \small{f(3)} \amp = \small{2 (3) = 6} \\ \small{f\left(\sqrt{7}\right)} \amp = \small{2 \left(\sqrt{7}\right) } \\ \end{align*}

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