Q: Which property is true for any one-to-one function $$f$$ and its inverse $$f^{-1}$$?

$$f^{-1}(f(x)) = x$ \text{ for every } x \text{ in the domain of } f$

Question 1

Which of the following best describes the domain and range of the inverse sine function $$y = \sin$$^{-1}$$ x$$?

Accepted Answer

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

Question 2

Find the exact value of $$\sin$$^{-1}$$\left(-1\right).$$

Accepted Answer

$$-\frac{\pi}{2}$$

Question 3

Which property is true for any one-to-one function $$f$$ and its inverse $$f^{-1}$$?

Accepted Answer

$$f^{-1}(f(x)) = x$ 	ext{ for every } x 	ext{ in the domain of } f$

Question 4

Find the exact value of $$\cos$$^{-1}$$\left(\frac{1}{2}\right).$$

Accepted Answer

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

Question 5

Which of the following is the correct range for the inverse cosine function $$y = \cos$$^{-1}$$ x$$?

Accepted Answer

$$0 \leq y \leq \pi$$

Question 6

Evaluate $$	an$$^{-1}$$(-\sqrt{3})$$ and select the correct answer.

Accepted Answer

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

Question 7

Which of the following composite functions is NOT defined?

Accepted Answer

$$\sin[\sin$$^{-1}$$(-2.5)]$$

Question 8

If $$f(x) = 2\sin x - 1$$ for $$-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$$, what is the inverse function $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \$$sin^{-1}$$\left(\frac{x+1}{2}\right)$$

Question 9

Solve for $$x$$: $$\sin$$^{-1}$$(x) = \frac{\pi}{6}.$$

Accepted Answer

$$x = \frac{1}{2}$$

Question 10

Which of the following statements about the graphs of a function and its inverse is correct?

Accepted Answer

They are symmetric with respect to the line $$y = x.$$

Question 11

Find the exact value of $$\sin\left(\sin$$^{-1}$$\left(\frac{1}{3}\right)\right).$$

Accepted Answer

$$\frac{1}{3}$$

Question 12

A medical device uses the function $$y = 	an$$^{-1}$$(x) to $$convert a sensor reading $$x to an $$angle $$y. If $$the sensor reads $$x = 1$$, what is the angle $$y in $$radians?

Accepted Answer

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

Question 13

Which of the following is the correct domain for the inverse tangent function $$y = 	an$$^{-1}$$ x$$?

Accepted Answer

$$-\infty \u003c x \u003c \infty$$

Question 14

Solve for $$x$$: $$2\sin x - 1 = 0$$, where $$-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}.$$

Accepted Answer

$$x = \frac{\pi}{6}$$

Question 15

Which of the following is true about the composite function $$\cos(\cos$$^{-1}$$(x))$$?

Accepted Answer

It equals $$x$$ for $$-1 \leq x \leq 1$$

Question 16

A satellite dish is oriented so that the angle of elevation $$	heta$$ satisfies $$	heta = \sin$$^{-1}$$(0.8). $$What is the value of $$	heta in $$radians, rounded to two decimal places?

Accepted Answer

$$0.93$$

Question 17

Find the exact value of $$\cos$$^{-1}$$\left(-\frac{\sqrt{3}}{2}\right).$$

Accepted Answer

$$\frac{5\pi}{6}$$

Question 18

Which of the following is the correct explicit equation for the inverse of $$y = 	an x$$ restricted to $$-\frac{\pi}{2} \u003c x \u003c \frac{\pi}{2}$$?

Accepted Answer

$$y = 	an$$^{-1}$$ x$$

Inverse Trigonometric Functions: Sine, Cosine, and Tangent

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