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Q.
If $f(x) = \cos^{-1} \left(\frac{2-|x|}{4}\right)+\log(3 - x)]^{-1}$, then its domain is given by
Inverse Trigonometric Functions
Solution:
The domain of $cos^{-1}\left(\frac{2-|x|}{4}\right) $ is given by
$-1 \,\leq\,\frac{2-|x|}{4}\,\leq\,\Rightarrow -4 \leq\,2-|x|\,\leq\,4$
$\Rightarrow \,-6\,\leq\,-|x|\,\leq\,2 \Rightarrow \,-2\,\leq\,|x|\leq\,6\,\Rightarrow \,|x|\leq\,6$
$\therefore $ domain of this function is -6 $\leq\,x\,\leq\,6$.
Also the domain of $\frac{1}{\log\,(3-x)}$ is given by 3-x > 0
and x $\neq$ 2 ($\because\,3-x \neq\,1)$
i,e., x= $\neq$ and x < 3
$\therefore $ required domain is $-6\,\le\,x < 3 $
and $x \,\neq\,2$ i.e., [-6, 2) U (2, 3)