Question

The domain of ${\cos }^{-1}\left(x^2-4\right)$ is
(a) [3, 5]
(b) [−1, 1]
(c) $\left[-\sqrt{5},-\sqrt{3}\right]\cup \left[\sqrt{3},\sqrt{5}\right]$
(d) $\left[-\sqrt{5},-\sqrt{3}\right]\cap \left[\sqrt{3},\sqrt{5}\right]$

Answer 1

The domain of ${\cos }^{-1}\left(x\right)$ is [−1, 1]
$$\begin{gathered} \therefore -1\le x^2-4\le 1 \\ \Rightarrow -1+4\le x^2-4+4\le 1+4 \\ \Rightarrow 3\le x^2\le 5 \\ \Rightarrow \pm \sqrt{3}\le x\le \pm \sqrt{5} \\ \Rightarrow x\in \left[-\sqrt{5},-\sqrt{3}\right]\cup \left[\sqrt{3},\sqrt{5}\right] \end{gathered}$$
Hence, the correct answer is option (c).