Question

The domain of the function ${\sin }^{-1}\left({\log }_2\left(\frac{x}{3}\right)\right)$

• A. $\left[0,6\right]$
• B. $[\frac{1}{2},3]$
• C. $[\frac{3}{2},6]$
• D. $[\frac{1}{2},2]$

Answer 1

The correct option is A $\left[0,6\right]$

We have,

${\sin }^{-1}\left[{\log }_2\left(\frac{x}{3}\right)\right]-1\le {\log }_2\left(\frac{x}{3}\right)\le 1-2\le \frac{x}{3}\le 2-6\le x\le \mathrm{6...........}\left(1\right)$

Since,

${\log }_2\left(\frac{x}{3}\right)\frac{x}{3}>\mathrm{0.............}\left(2\right)$

$combition\left(1\right)\&\left(2\right)\Rightarrow 0<x<6\Rightarrow \left[0,6\right]$

Hence, this is the answer.