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

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

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?

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

Q: Suppose $f(x)$ models the temperature (in $^\circ$C) 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?

$\dfrac{25 - 10}{5 - 2} = 5$ $^\circ$C$/$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?

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

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 < 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

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

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 $^\circ$C) 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

$\dfrac{25 - 10}{5 - 2} = 5$ $^\circ$C$/$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

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

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 > 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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