Question

Find the domain of definition of $f\left(x\right)={\cos }^{-1}\left(x^2-4\right)$.

Answer 1

For ${\cos }^{-1}\left(x^2-4\right)$ to be defined
$$\begin{gathered} -1\le x^2-4\le 1 \\ \Rightarrow 3\le x^2\le 5 \\ \Rightarrow x\in \left[-\sqrt{5},-\sqrt{3}\right]\cup \left[\sqrt{3},\sqrt{5}\right] \end{gathered}$$
Hence, the domain of $f\left(x\right)\mathrm{is}\left[-\sqrt{5},-\sqrt{3}\right]\cup \left[\sqrt{3},\sqrt{5}\right]$.