Question

Domain of $f\left(x\right)={\sin }^{-1}\left({\log }_2\frac{x}{2}\right)$ is

Answer 1

We know that ${\sin }^{-1}x$ exists if $-1\le x\le 1$

$\therefore f\left(x\right)={\sin }^{-1}\left[lo{g}_2\frac{x}{2}\right]$ exists if $-1\le lo{g}_2\frac{x}{2}\le 1$

Taking base $2$ to remove $lo{g}_2$

$2^{-1}\le \frac{x}{2}\le 2$

$\frac{1}{2}\le \frac{x}{2}\le 2$

$1\le x\le 4$

Multiplying by $2$ or

Hence the domain of $f\left(x\right)$ is

$[-4,-1]\cup [1,4]$.