Question

The domain of $f(x) = sin^{-1} [Log_2 (\frac {x} {2})]$ is

• A. $0 \leq x \leq 1$
• B. $0 \leq x \leq 4$
• C. $1 \leq x \leq 4$
• D. $4 \leq x \leq 6$

Answer 1

Answer: (C)

$1 \leq x \leq 4$

Answer 2

Given function, $f(x)=\sin ^{-1}\left[\log _{2}\left(\frac{x}{2}\right)\right]$
Since, $-1 \leq \sin x \leq-1$
$\Rightarrow -1 \leq \log _{2}\left(\frac{x}{2}\right) \leq 1$
$\Rightarrow 2^{-1} \leq x / 2 \leq 2^{1}$
$\Rightarrow 2^{0} \leq x \leq 2^{2}$
$\Rightarrow 1 \leq x \leq 4$
Required domain is $x \in[1,4].$