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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