Question

Domain of $f\left(x\right)={\sin }^{-1}\left[{\log }_2\left(\frac{x^2}{2}\right)\right]$, where $[.]$ denotes greatest integer functions, is

• A. $[-\sqrt{8,}\sqrt{8}]$
• B. $[-\sqrt{8,}-1]\cup (1,\sqrt{8}]$
• C. $(-2,-1)\cup (1,2)$
• D. None of these

Answer 1

The correct option is C $(-2,-1)\cup (1,2)$

$f\left(x\right)si{n}^{-1}\left[lo{g}_2\right(\frac{x^2}{2}\left)\right]$

For $si{n}^{1}x,x$ must lie strietly in the range [-1,1]

$\Rightarrow -1\le lo{g}_2\frac{x^2}{2}\le 1$

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

$\Rightarrow 1\le x^2\le 4$

$\Rightarrow 1\le x\le 2$ or $-2\le x\le -1$

$\therefore$ Domain of f is $[-2,-1]\cup [1,2]$