Question

$f\left(x\right)={\sin }^{-1}\left[{\log }_2\left(\frac{x^2}{2}\right)\right]$ where $[.]$ denotes the greatest integer function.

Answer 1

$\begin{array}{cc}f\left(x\right)={\sin }^{-1}\left[{\log }_2\left(\frac{x^2}{2}\right)\right]& \\ \Rightarrow -1\le \left[{\log }_2\left(\frac{x^2}{2}\right)\right]\le 1& \\ \Rightarrow 2^{-1}\le \frac{x^2}{2}\le 2& \left[domainof{\sin }^{-1}=\left[-1,1\right]\right]\\ \Rightarrow \frac{1}{2}\le \frac{x^2}{2}\le 2& \\ \Rightarrow 1\le x^2\le 4& \\ 1\le x\le 2& \end{array}$