The domain of
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).

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