Domain and range of $f(x)=3\sin^{-1}(5x)$ I am trying to determine the domain of $$f(x)=3\sin^{-1}(5x).$$ Since the domain of $g(x)=\sin^{-1}(x)$ is given by the set $\{x\in\mathbb{R}:-1\leq x\leq 1\}$, I understand that the domain of $f$ must satisfy
\begin{align}
-1&\leq5x\leq1 \\
-\frac{1}{5}&\leq x\leq\frac{1}{5}.
\end{align}
I know the range of $g$ is given by the set $\left\{y\in\mathbb{R}:-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\right\}$, but I am unable to determine the range of $f$. Any suggestions are helpful.

Yes, the domain of $f$ is given for all $x\in [-1/5,1/5]$ as you said, that's perfect work.  Now, since as you said the range of $x\mapsto \sin^{-1}x$ is given by $[-\pi/2,\pi/2]$ and the factor $3$ affects the range so the range of $x\mapsto 3\sin^{-1}(5x)$ is given by $y\in [-3\pi/2,3\pi/2]$.
