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 $$-10 \u003c x \\leq 2$$, $$2 \u003c x \\leq 8$$, and $$8 \u003c x \\leq 16$$?

$$f(x) = \\begin{cases} -6, & -10 \u003c x \\leq 2 \\ 2x, & 2 \u003c x \\leq 8 \\ 2x - 8, & 8 \u003c x \\leq 16 \\end{cases}$$

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

$$f(x) = \\begin{cases} 2x - 12, & x \\geq 6 \\ 12 - 2x, & x \u003c 6 \\end{cases}$$

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 $$0 \u003c x \\leq 6$$, $$30$$ for $$6 \u003c x \\leq 12$$, $$40$$ for $$12 \u003c x \\leq 18$$, and $$50$$ for $$18 \u003c x \\leq 24$$, what is the domain and range of this function?

Domain: $$0 \u003c x \\leq 24$$, 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?

$$E(h) = \\begin{cases} 10h, & 0 \\leq h \\leq 40 \\ 15h - 200, & h \u003e 40 \\end{cases}$$

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 $$-10 \u003c x \leq 2$$, $$2 \u003c x \leq 8$$, and $$8 \u003c x \leq 16$$?

Accepted Answer

$$f(x) = \begin{cases} -6, & -10 \u003c x \leq 2 \ 2x, & 2 \u003c x \leq 8 \ 2x - 8, & 8 \u003c x \leq 16 \end{cases}$$

Question 2

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

Accepted Answer

$$f(x) = \begin{cases} 2x - 12, & x \geq 6 \ 12 - 2x, & x \u003c 6 \end{cases}$$

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 $$0 \u003c x \leq 6$$, $$30$$ for $$6 \u003c x \leq 12$$, $$40$$ for $$12 \u003c x \leq 18$$, and $$50$$ for $$18 \u003c x \leq 24$$, what is the domain and range of this function?

Accepted Answer

Domain: $$0 \u003c x \leq 24$$, 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

$$E(h) = \begin{cases} 10h, & 0 \leq h \leq 40 \ 15h - 200, & h \u003e 40 \end{cases}$$

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 $$f(x) = \begin{cases} -6, & -5 \leq x \u003c 1 \ 2x, & 1 \leq x \u003c 8 \ 2x - 8, & 8 \leq x \leq 16 \end{cases}$$, 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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