Chapter 4, Problem 394RE

For the following exercises, determine whether the statement is true or false. Justify your answer with a proof or a counterexample.

394. The domain of $f\left(x,y\right)=x^3\sin -1$ is $x=$ all real numbers, and $-\pi \le y\le \pi$.

To determine

Whether the given statement is true or false, with counterexample.

Answer to Problem 394RE

Thus, the given statement is false.

Explanation of Solution

Given:

Domain of $f\left(x,y\right)=x^3{\sin }^{-1}\left(y\right),x\in R,-\pi \le y\le \pi$

Calculation:

As here, $f\left(x,y\right)=x^3{\sin }^{-1}\left(y\right),x\in R,-\pi \le y\le \pi$

$x^3$ is continuous for all real values, thus its domain is all real values, but the domain of the function ${\sin }^{-1}\left(y\right)$ is $-\frac{\pi }{2}\le y\le \frac{\pi }{2}$

Thus, the given statement is false.