Question

The domain of the function $f ( x )=\sqrt{\log _{1 / 2}\left(\log _5\left(\left[ x ^2\right]-3\right)\right)}$ is
[Note: Where $[ k ]$ denotes greatest integer function less than or equal to $k$ ]

• A. $[-3,-2] \cup[2,3]$
• B. $(-3,-2] \cup[2,3)$
• C. $(-3,-\sqrt{5}] \cup[\sqrt{5}, 3)$
• D. $(-2 \sqrt{2},-2) \cup(2,2 \sqrt{2})$

Answer 1

Answer: (C)

$f ( x )=\sqrt{\log _{1 / 2} \log _5\left(\left[ x ^2\right]-3\right)} $
$0 \leq \log _{1 / 2} \log _5\left(\left[ x ^2\right]-3\right)$
$\Rightarrow 0<\log _5\left(\left[ x ^2\right]-3\right) \leq 1$
$\Rightarrow 1<\left[x^2\right]-3 \leq 5$
$4<\left[ x ^2\right] \leq 8$
$5 \leq x ^2<9$
$x \in(-3,-\sqrt{5}] \cup[\sqrt{5}, 3) .$