Question 1

Which of the following conditions must a function $f: A 	o B$ satisfy in order for its inverse $f^{-1}$ to exist?

Accepted Answer

It must be a one-to-one (injective) function.

Question 2

Given $f(x) = x^2$ with domain $\mathbb{R}$, which of the following best describes the range of $f$?

Accepted Answer

$\{ y \mid y \geq 0 \}$

Question 3

Why does the function $f(x) = x^2$ not have an inverse on $\mathbb{R}$?

Accepted Answer

Because it is not one-to-one on $\mathbb{R}$.

Question 4

For which interval is the inverse $f^{-1}(x)$ of $f(x) = x^2$ properly defined?

Accepted Answer

$[0, \infty)$$ $or$ $(-\infty, 0]$ $but$ $not$ $\mathbb{R}$

Question 5

Which of the following is the correct domain and range for the inverse sine function $y = \sin^{-1}(x)$?

Accepted Answer

Domain: $-1 \leq x \leq 1$, Range: $-\frac{\pi}{2} \leq y \leq \frac{\pi}{2}$

Question 6

Evaluate $\sin^{-1}\left(-\frac{1}{2}\right)$.

Accepted Answer

$-\frac{\pi}{6}$

Question 7

Evaluate $\cos^{-1}\left(-\frac{1}{2}\right)$.

Accepted Answer

$\frac{2\pi}{3}$

Question 8

Evaluate $	an^{-1}(-\sqrt{3})$.

Accepted Answer

$-\frac{\pi}{3}$

Question 9

Which of the following is true about $\sin^{-1}(\sin(\frac{5\pi}{8}))$?

Accepted Answer

$\sin^{-1}(\sin(\frac{5\pi}{8})) = \frac{3\pi}{8}$

Question 10

For which values of $x$ is $\cos^{-1}(x)$ defined?

Accepted Answer

$-1 \leq x \leq 1$

Question 11

What is the value of $\cos(\cos^{-1}(-2.4))$?

Accepted Answer

Not defined

Question 12

What is the value of $\cos(\cos^{-1}(\pi))$?

Accepted Answer

Not defined

Question 13

Evaluate $\cos^{-1}(\cos(\frac{2\pi}{3}))$.

Accepted Answer

$\frac{2\pi}{3}$

Question 14

Which of the following statements about the inverse trigonometric functions is correct?

Accepted Answer

$\sin^{-1}(x)$ is only defined for $-1 \leq x \leq 1$ and returns values in $[-\frac{\pi}{2}, \frac{\pi}{2}]$

Question 15

If $f(x) = x^2$ and $f^{-1}$ is defined only for $x \geq 0$, what is $f^{-1}(y)$?

Accepted Answer

$\sqrt{y}$

Question 16

Suppose a chemical reaction's rate $R$ is modeled by $R = \sin(	heta)$, where $	heta$ is the angle of a rotating paddle in a mixing tank. If the measured rate is $R = 0.5$, what is the principal value of $	heta$?

Accepted Answer

$\frac{\pi}{6}$

Question 17

A medical device uses the function $f(x) = 	an(x)$ to convert an input signal $x$ to an output. If the output is $-1$, what is the principal value of the input $x$?

Accepted Answer

$-\frac{\pi}{4}$

Question 18

A satellite's position is modeled by $y = \cos^{-1}(x)$, where $x$ is the normalized altitude. If $x = 0$, what is the satellite's position $y$?

Accepted Answer

$\frac{\pi}{2}$

Question 19

If $f(x)$ is a one-to-one function and $f(a) = b$, which of the following is true about its inverse?

Accepted Answer

$f^{-1}(b) = a$

Inverse Functions and Inverse Trigonometric Functions in Precalculus

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