Q: Given the graph of a piecewise-defined function with three distinct linear segments, which of the following correctly expresses $f(x)$ for the intervals $-10 < x \\leq 2$, $2 < x \\leq 8$, and $8 < x \\leq 16$?

$f(x) = \\begin{cases} -6, & -10 < x \\leq 2 \\ 2x, & 2 < x \\leq 8 \\ 2x - 8, & 8 < x \\leq 16 \\end{cases}$

Q: Which of the following expresses $f(x) = |2x - 12|$ as a piecewise-defined function?

$f(x) = \\begin{cases} 2x - 12, & x \\geq 6 \\ 12 - 2x, & x < 6 \\end{cases}$

Q: A step function models the cost $y$ of renting a bicycle for $x$ hours, where $y$ increases at certain intervals. If the cost is $20$ for $0 < x \\leq 6$, $30$ for $6 < x \\leq 12$, $40$ for $12 < x \\leq 18$, and $50$ for $18 < x \\leq 24$, what is the domain and range of this function?

Domain: $0 < x \\leq 24$, Range: $\\{20, 30, 40, 50\\}$

Q: Gordon earns $10$ per hour for up to $40$ hours per week as a tour guide. For hours over $40$, he earns $15$ per hour. Which piecewise-defined function models his weekly earnings $E(h)$ based on hours $h$ worked?

$E(h) = \\begin{cases} 10h, & 0 \\leq h \\leq 40 \\ 15h - 200, & h > 40 \\end{cases}$

Q: A student claims that the function $f(x) = |2x - 12|$ is always positive. Is this correct?

No, $f(x)$ is zero when $x = 6$.

Q: Which graph best represents the function $f(x) = |2x - 12|$?

A V-shaped graph with vertex at $(6, 0)$

Q: A piecewise-defined function is used to model a real-world scenario. Which of the following is a valid reason for using such a function?

The scenario involves different rules or rates for different intervals.

Q: Which of the following is true about the graph of a piecewise-defined function?

It may have discontinuities at the boundaries between pieces.

Question 1

Given the graph of a piecewise-defined function with three distinct linear segments, which of the following correctly expresses $f(x)$ for the intervals $-10 < x \leq 2$, $2 < x \leq 8$, and $8 < x \leq 16$?

Accepted Answer

$f(x) = \begin{cases} -6, & -10 < x \leq 2 \ 2x, & 2 < x \leq 8 \ 2x - 8, & 8 < x \leq 16 \end{cases}$

Question 2

Which of the following expresses $f(x) = |2x - 12|$ as a piecewise-defined function?

Accepted Answer

$f(x) = \begin{cases} 2x - 12, & x \geq 6 \ 12 - 2x, & x < 6 \end{cases}$

Question 3

A step function models the cost $y$ of renting a bicycle for $x$ hours, where $y$ increases at certain intervals. If the cost is $20$ for $0 < x \leq 6$, $30$ for $6 < x \leq 12$, $40$ for $12 < x \leq 18$, and $50$ for $18 < x \leq 24$, what is the domain and range of this function?

Accepted Answer

Domain: $0 < x \leq 24$, Range: $\{20, 30, 40, 50\}$

Question 4

Gordon earns $10$ per hour for up to $40$ hours per week as a tour guide. For hours over $40$, he earns $15$ per hour. Which piecewise-defined function models his weekly earnings $E(h)$ based on hours $h$ worked?

Accepted Answer

$E(h) = \begin{cases} 10h, & 0 \leq h \leq 40 \ 15h - 200, & h > 40 \end{cases}$

Question 5

If $f(x) = |2x - 12|$, what is the minimum value of $f(x)$ and at what $x$ does it occur?

Accepted Answer

Minimum value is $0$ at $x = 6$

Question 6

Which of the following is NOT a characteristic of a step function?

Accepted Answer

It is continuous over its entire domain.

Question 7

Given the piecewise function $f(x) = \begin{cases} -6, & -5 \leq x < 1 \ 2x, & 1 \leq x < 8 \ 2x - 8, & 8 \leq x \leq 16 \end{cases}$, what is $f(10)$?

Accepted Answer

$12$

Question 8

A student claims that the function $f(x) = |2x - 12|$ is always positive. Is this correct?

Accepted Answer

No, $f(x)$ is zero when $x = 6$.

Question 9

If Gordon works $45$ hours in a week, what are his total earnings according to the piecewise function $E(h)$?

Accepted Answer

$275$

Question 10

Which graph best represents the function $f(x) = |2x - 12|$?

Accepted Answer

A V-shaped graph with vertex at $(6, 0)$

Question 11

For the step function modeling bike rental cost, what is the cost for renting a bike for $15$ hours?

Accepted Answer

$40$

Question 12

Which of the following intervals is NOT included in the domain of the bike rental cost function?

Accepted Answer

$x = 25$

Question 13

If $f(x) = |2x - 12|$, for which values of $x$ is $f(x) = 8$?

Accepted Answer

$x = 2$ and $x = 10$

Question 14

Which of the following best describes the range of the function $f(x) = |2x - 12|$ for $x$ in $[0, 10]$?

Accepted Answer

$[0, 12]$

Question 15

A piecewise-defined function is used to model a real-world scenario. Which of the following is a valid reason for using such a function?

Accepted Answer

The scenario involves different rules or rates for different intervals.

Question 16

If $f(x)$ is a step function with values $20$, $30$, $40$, and $50$ for intervals of $x$, what is the minimum and maximum value of $f(x)$?

Accepted Answer

Minimum: $20$, Maximum: $50$

Question 17

Which of the following is true about the graph of a piecewise-defined function?

Accepted Answer

It may have discontinuities at the boundaries between pieces.

Question 18

If $f(x) = |2x - 12|$, what is $f(0)$?

Accepted Answer

$12$

Question 19

Which of the following best describes the graph of Gordon's earnings function $E(h)$?

Accepted Answer

A linear graph with a change in slope at $h = 40$

Piecewise-Defined Functions: Concepts, Graphs, and Applications

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