The domain of $\mathrm{f}\left(\mathrm{x}\right)={\cos }^{-1}\left(\frac{2-\left|\mathrm{x}\right|}{4}\right)+\left(\log (3-\mathrm{x})\right){ }^{-1}$ is

${\cos }^{-1}\left(\frac{2-\left|\mathrm{x}\right|}{4}\right)$ exists if
$-1\le \frac{2-\left|\mathrm{x}\right|}{4}\le 1\text{ and }3-\mathrm{x}>0\text{ and }3-\mathrm{x}\ne 1$
Thus, the domain of the given function is
$\{\mathrm{x}\mid -6\le \mathrm{x}\le 6\}\cap \{\mathrm{x}\mid \mathrm{x}\ne 2,\mathrm{x}<3\}=[-6,2)\cup (2,3)$