The domain of the function $\mathrm{f}\left(\mathrm{x}\right)=\sqrt{{\cos }^{-1}\left(\frac{1-\left|\mathrm{x}\right|}{2}\right)} \mathrm{is}$

$$\begin{gathered} {\cos }^{-1}\left(\frac{1-\left|\mathrm{x}\right|}{2}\right)\text{ is defined if }-1\le \frac{1-\left|\mathrm{x}\right|}{2}\le 1 \\ \begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-2\le 1-\left|\mathrm{x}\right|\le 2\Rightarrow -3\le -\left|\mathrm{x}\right|\le 1\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-1\le \left|\mathrm{x}\right|\le 3\end{array} \end{gathered}$$
|x| ≥ – 1 is true for all real values of x.
$$\begin{gathered} \text{ Also }\sqrt{{\cos }^{-1}\left(\frac{1-\left|\mathrm{x}\right|}{2}\right)}\text{ is defined if }{\cos }^{-1}\left(\frac{1-\left|\mathrm{x}\right|}{2}\right)\ge 0 \\ \begin{array}{l}\Rightarrow \frac{1-\left|\mathrm{x}\right|}{2}\ge \cos 0=1\\ \Rightarrow 1-\left|\mathrm{x}\right|\ge 2\\ \Rightarrow \left|\mathrm{x}\right|\le -1\text{ (Absurd) }\end{array} \end{gathered}$$
Domain ( f ) = [– 3, 3]