Question

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

• A. $\frac{1}{2},3$
• B. $\frac{1}{2},4$
• C. $\frac{3}{2},6$
• D. $\frac{1}{2},2$

Answer 1

The correct option is C $\frac{3}{2},6$

$y={\sin }^{-1}\left({\log }_2\left(\frac{x}{3}\right)\right)$

$y={\sin }^{-1}\theta$ is defined for $\theta \in [-1,1]$

${\log }_2\left(\frac{x}{3}\right)\in [-1,1]$

$\Rightarrow {\log }_2\frac{x}{3}\ge -1$ and $\Rightarrow {\log }_2\frac{x}{3}\le 1$

$\Rightarrow \frac{x}{3}\ge \frac{1}{2}$ and $\frac{x}{3}\le 2$

$x\ge \frac{3}{2}$ and $x\le 6$

$\therefore x\in [\frac{3}{2},6]$