Q: Which of the following statements best describes when a function $f(x)$ has an inverse?

If $f(x)$ is one-to-one, meaning $f(x_1) \ne f(x_2)$ whenever $x_1 \ne x_2$.

Question 1

Which of the following statements best describes when a function $f(x)$ has an inverse?

Accepted Answer

If $f(x)$ is one-to-one, meaning $f(x_1) 
e f(x_2)$ whenever $x_1 
e x_2$.

Question 2

Given $f(x) = 3x + 2$, what is the inverse function $f^{-1}(y)$?

Accepted Answer

$f^{-1}(y) = \frac{y-2}{3}$

Question 3

If a particle's velocity is given by $v(t) = 3t + 2$, at what time $t$ is the velocity $17$ m/s?

Accepted Answer

$t = 5$

Question 4

What is the domain of the inverse cosine function $\cos^{-1}(x)$?

Accepted Answer

$[-1, 1]$

Question 5

What is $\cos^{-1}\left(\frac{1}{2}\right)$?

Accepted Answer

$\frac{\pi}{3}$

Question 6

What is $\cos(\sin^{-1}(1))$?

Accepted Answer

$0$

Question 7

Suppose $f(x) = \sqrt[4]{x} - 2$. What is the domain of $f(x)$?

Accepted Answer

$[0, \infty)$

Question 8

Suppose $f(x) = \sqrt[4]{x} - 2$. What is the range of $f(x)$?

Accepted Answer

$[-2, \infty)$

Question 9

Given $f(x) = \sqrt[4]{x} - 2$, what is the domain of its inverse function?

Accepted Answer

$[-2, \infty)$

Question 10

Given $f(x) = \sqrt[4]{x} - 2$, what is the range of its inverse function?

Accepted Answer

$[0, \infty)$

Question 11

Which of the following is NOT a property of an inverse function $f^{-1}(y)$ for a one-to-one function $f(x)$?

Accepted Answer

$f^{-1}(x)$ is always defined for all real numbers

Question 12

A real-world scenario: The temperature $T$ in Celsius is related to Fahrenheit $F$ by $F = \frac{9}{5}T + 32$. What is the inverse function to convert Fahrenheit to Celsius?

Accepted Answer

$T = \frac{5}{9}(F - 32)$

Question 13

Which of the following functions is NOT one-to-one?

Accepted Answer

$f(x) = x^2$

Question 14

If $f(x)$ is one-to-one and $f(3) = 7$, what is $f^{-1}(7)$?

Accepted Answer

$3$

Question 15

Which of the following is the correct process to find the inverse of a function $f(x)$?

Accepted Answer

Replace $f(x)$ with $y$, solve for $x$ in terms of $y$, then swap $x$ and $y$.

Question 16

If $f(x) = 2x - 5$ and $g(x) = f^{-1}(x)$, what is $g(9)$?

Accepted Answer

$7$

Inverse Functions and One-to-One Functions: Definitions, Properties, and Examples

Study Guide - Practice Questions

Study Guide - Flashcards