Question:
The domain of $\cos ^{-1}\left(x^{2}-4\right)$ is
(a) $[3,5]$
(b) $[-1,1]$
(c) $[-\sqrt{5},-\sqrt{3}] \cup[\sqrt{3}, \sqrt{5}]$
(d) $[-\sqrt{5},-\sqrt{3}] \cap[\sqrt{3}, \sqrt{5}]$
Solution:
The domain of $\cos ^{-1}(x)$ is $[-1,1]$
$\therefore-1 \leq x^{2}-4 \leq 1$
$\Rightarrow-1+4 \leq x^{2}-4+4 \leq 1+4$
$\Rightarrow 3 \leq x^{2} \leq 5$
$\Rightarrow \pm \sqrt{3} \leq x \leq \pm \sqrt{5}$
$\Rightarrow x \in[-\sqrt{5},-\sqrt{3}] \cup[\sqrt{3}, \sqrt{5}]$
Hence, the correct answer is option (c).