Chapter 7.1, Problem 10AYU

To determine

To define: Whether the statement “The domain of $y={\sin }^{-1}x$ is $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$ .” is true or false.

Answer to Problem 10AYU

The statement “The domain of $y={\sin }^{-1}x$ is $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$ .” is false.

Explanation of Solution

Given information:

The statement “The domain of $y={\sin }^{-1}x$ is $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$ .”

Consider the provided statement “The domain of $y={\sin }^{-1}x$ is $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$ .”

Recall that the range of function $f$ equals the domain of it’s inverse, $f^{-1}$ .

Range of sine function is $\left[-1,1\right]$ .

According to the statement, domain of the inverse sine function is $\left[-1,1\right]$ .

Thus, the statement “The domain of $y={\sin }^{-1}x$ is $-\frac{\pi }{2}\le x\le \frac{\pi }{2}$ .” is false.