Find the  range of f(x) $f\left(x\right)={\sin }^{-1}\left[x\right]$

$f\left(x\right)={\sin }^{-1}\left[x\right]\text{ is defined if }$

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\le \left[x\right]\le 1\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\left[x\right]=-1,0,1\\ \text{ or }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\in [-1,2)\end{array}$

$\text{ So, range is }\left\{{\sin }^{-1}(-1),{\sin }^{-1}0,{\sin }^{-1}1\right\}\equiv \{-\pi /2,0,\pi /2\}$