First slide

Functions (XII)

Question

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

Moderate

A

$[- 2, 6]$
B

$[- 6, 2) \cup (2, 3)$
C

$[- 6, 2]$
D

$[- 2, 2] \cup (2, 3)$

Solution

$\begin{array}{l} \cos^{- 1} (\frac{2 - |x|}{4}) exists if \\ - 1 \leq \frac{2 - |x|}{4} \leq 1 \\ or - 6 \leq - |x| \leq 2 \\ or - 2 \leq |x| \leq 6 \\ or |x| \leq 6 \\ or - 6 \leq x \leq 6 \end{array}$
The function ${[\log (3 - x)]}^{- 1} = \frac{1}{\log (3 - x)}$ is defined if $3 - x > 0$
and $x \neq 2$ i.e., if $x \neq 2$ and x<3.
Thus, the domain of the given function is
$\{x | - 6 \leq x \leq 6\} \cap \{x | x \neq 2, x < 3\} = [- 6, 2) \cup (2, 3)$

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