Q: Given $f(x) = \frac{1}{x}$, which of the following statements is true about its domain?

The domain is all real numbers except $x = 0$.

Q: A function $f$ is defined as $f(x) = \begin{cases} 2x & x < 0 \\ x^2 & x \geq 0 \end{cases}$. What is $f(-3) + f(2)$?

$f(-3) = 2(-3) = -6$, $f(2) = 2^2 = 4$, so $-6 + 4 = -2$

Q: Which of the following best describes the graph of $y = |x|$?

A V-shaped graph opening upwards.

Q: Which of the following is true about the function $y = x^3$?

It is an odd function and its graph is symmetric about the origin.

Q: If $f(x) = |x|$ and $g(x) = x^2$, what is the difference quotient $\frac{f(x+h) - f(x)}{h}$ when $x = -2$ and $h = 1$?

$\frac{|(-2)+1| - |-2|}{1} = \frac{|-1| - 2}{1} = -1$

Question 1

Which of the following is the correct definition of a piecewise function?

Accepted Answer

A function that is defined by two or more equations over a specified domain.

Question 2

Given the function $f(x) = \begin{cases} 3x + 5 & 	ext{if } -4 \leq x < 1 \ 4x - 2 & 	ext{if } x \geq 1 \end{cases}$, what is $f(-2)$?

Accepted Answer

$f(-2) = 3(-2) + 5 = -1$

Question 3

Which of the following is the parent function for the absolute value?

Accepted Answer

$y = |x|$

Question 4

What is the domain of the function $f(x) = \sqrt{x}$?

Accepted Answer

$[0, \infty)$

Question 5

Given $f(x) = \frac{1}{x}$, which of the following statements is true about its domain?

Accepted Answer

The domain is all real numbers except $x = 0$.

Question 6

Evaluate $f(0)$ for the piecewise function $f(x) = \begin{cases} 3x + 1 & x < 1 \ -2x + 1 & x \geq 1 \end{cases}$.

Accepted Answer

$f(0) = 3(0) + 1 = 1$

Question 7

Which of the following functions is NOT a parent function listed in the library of functions?

Accepted Answer

$y = x^4$

Question 8

A function $f$ is defined as $f(x) = \begin{cases} 2x & x < 0 \ x^2 & x \geq 0 \end{cases}$. What is $f(-3) + f(2)$?

Accepted Answer

$f(-3) = 2(-3) = -6$, $f(2) = 2^2 = 4$, so $-6 + 4 = -2$

Question 9

Which of the following best describes the graph of $y = |x|$?

Accepted Answer

A V-shaped graph opening upwards.

Question 10

For the function $f(x) = \begin{cases} 3x + 5 & -4 \leq x < 1 \ 4x - 2 & x \geq 1 \end{cases}$, what is the value of $f(1)$?

Accepted Answer

$f(1) = 4(1) - 2 = 2$

Question 11

Which of the following is the range of the quadratic parent function $y = x^2$?

Accepted Answer

$[0, \infty)$

Question 12

A medical dosage is modeled by the function $f(x) = \begin{cases} 2x + 10 & x < 5 \ 3x - 5 & x \geq 5 \end{cases}$, where $x$ is the number of hours after administration. What is the dosage at $x = 6$ hours?

Accepted Answer

$f(6) = 3(6) - 5 = 13$

Question 13

Which of the following is true about the function $y = x^3$?

Accepted Answer

It is an odd function and its graph is symmetric about the origin.

Question 14

A piecewise function is given by $f(x) = \begin{cases} x^2 & x < 2 \ 2x + 1 & x \geq 2 \end{cases}$. What is $f(2)$?

Accepted Answer

$f(2) = 2(2) + 1 = 5$

Question 15

Which function from the library of functions has a domain of all real numbers except $x = 0$ and a range of all real numbers except $y = 0$?

Accepted Answer

$y = \frac{1}{x}$

Question 16

If $f(x) = |x|$ and $g(x) = x^2$, what is the difference quotient $\frac{f(x+h) - f(x)}{h}$ when $x = -2$ and $h = 1$?

Accepted Answer

$\frac{|(-2)+1| - |-2|}{1} = \frac{|-1| - 2}{1} = -1$

Question 17

Which of the following is the correct domain for the piecewise function $f(x) = \begin{cases} 3x + 5 & -4 \leq x < 1 \ 4x - 2 & x \geq 1 \end{cases}$?

Accepted Answer

$[-4, \infty)$

Question 18

A chemical reaction rate is modeled by $f(x) = \sqrt{x}$, where $x$ is the concentration in mol/L. If the concentration is $9$ mol/L, what is the reaction rate?

Accepted Answer

$f(9) = \sqrt{9} = 3$

Library of Functions and Piecewise-defined Functions: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards