Q: A taxi company charges $$\3$ for the first mile and $$\$$2$$ for each additional mile. Which of the following is the correct piecewise function for the cost $$C(x) of a $$ride of $$x$$ miles?

$$C(x) = \begin{cases} 3 & x \leq 1 \\ 3 + 2(x-1) & x \u003e 1 \end{cases}$$

Q: Suppose $$f(x)$$ models the temperature (in $$^\circC) at $$time $$x (in $$hours). If $$f(2) = 10$$ and $$f(5) = 25$$, what is the average rate of change of temperature from hour 2 to hour 5?

$$\dfrac{25 - 10}{5 - 2} = 5$$ $$^\circC$/$$hour

Q: A cell phone plan charges $$\20$ for up to 2 GB of data and $$\$$10$$ for each additional GB. Which piecewise function models the total cost $$C(x)$$ for $$x GB of $$data?

$$C(x) = \begin{cases} 20 & x \leq 2 \\ 20 + 10(x-2) & x \u003e 2 \end{cases}$$

Question 1

Which of the following best describes a piecewise function?

Accepted Answer

A function that is defined by two or more equations over different parts of its domain.

Question 2

Given the piecewise function: 
$$
 f(x) = \begin{cases} x^2$ & 	ext{if } x \u003c 0 \ 2x + 1 & 	ext{if } x \geq 0 \end{cases} 
$$
What is $$f(-2)$$?

Accepted Answer

$$f(-2) = (-2)^2$ = 4$$

Question 3

What is the average rate of change of the function $$f(x) = x^2$ from x = 1$ to x = 4$?

Accepted Answer

$$\dfrac{f(4) - f(1)}{4 - 1} = \dfrac{16 - 1}{3} = 5$$

Question 4

A taxi company charges $$\3$ for the first mile and $$\$$2$$ for each additional mile. Which of the following is the correct piecewise function for the cost $$C(x) of a $$ride of $$x$$ miles?

Accepted Answer

$$C(x) = \begin{cases} 3 & x \leq 1 \ 3 + 2(x-1) & x \u003e 1 \end{cases}$$

Question 5

If $$f(x) is a $$function and the average rate of change from $$x = a to$$ $$x = b is $$zero, what does this imply about $$f(a)$$ and $$f(b)$$?

Accepted Answer

$$f(a) = f(b)$$

Question 6

Given the graph in the image, what is the formula for the average rate of change between $$x = a$$ and $$x = b$$?

Accepted Answer

$$\dfrac{f(b) - f(a)}{b - a}$$

Question 7

Suppose $$f(x)$$ models the temperature (in $$^\circC) at $$time $$x (in $$hours). If $$f(2) = 10$$ and $$f(5) = 25$$, what is the average rate of change of temperature from hour 2 to hour 5?

Accepted Answer

$$\dfrac{25 - 10}{5 - 2} = 5$$ $$^\circC$/$$hour

Question 8

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

Accepted Answer

It is always continuous everywhere.

Question 9

A cell phone plan charges $$\20$ for up to 2 GB of data and $$\$$10$$ for each additional GB. Which piecewise function models the total cost $$C(x)$$ for $$x GB of $$data?

Accepted Answer

$$C(x) = \begin{cases} 20 & x \leq 2 \ 20 + 10(x-2) & x \u003e 2 \end{cases}$$

Question 10

If $$f(x) = |x|$$, what is the average rate of change from $$x = -3 to$$ $$x = 3$$?

Accepted Answer

$$\dfrac{f(3) - f(-3)}{3 - (-3)} = \dfrac{3 - 3}{6} = 0$$

Question 11

A company’s revenue $$R(x) in $$thousands of dollars is given by the piecewise function:
$$
R(x) = \begin{cases} 5x & 0 \leq x \leq 10 \ 50 + 3(x-10) & x \u003e 10 \end{cases}
$$
What is the average rate of change from $$x = 8 to$$ $$x = 12$$?

Accepted Answer

$$\dfrac{R(12) - R(8)}{12 - 8} = \dfrac{56 - 40}{4} = 4$$

Question 12

Which real-world scenario is best modeled by a piecewise function?

Accepted Answer

A car rental company charges a flat fee for the first day and a different rate for each additional day.

Piecewise-Defined Functions and Average Rate of Change

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