Question

If $f(x)=\sqrt{3|x|-x-2}$ and $g(x)=\sin x$, then domain of $f \circ g(x)$ is

• A. $\left\{2 n \pi+\frac{\pi}{2}\right\}$, where $n \in I$
• B. $\left[2 n \pi+\frac{7 \pi}{6}, 2 n \pi+\frac{11 \pi}{6}\right]$, where $n \in I$
• C. $\left\{2 n \pi+\frac{\pi}{6}\right\}$, where $n \in I$
• D. $\left\{(4 m+1) \frac{\pi}{2} ; m \in I\right\} \cup\left[2 n \pi+\frac{7 \pi}{6}, 2 n \pi+\frac{11 \pi}{6}\right]$, where $m, n \in I$

Answer 1

Answer: (D)

$f(g(x))=\sqrt{3|\sin x|-\sin x-2} $
$\therefore 3|\sin x|-\sin x-2 \geq 0$
$\sin x>0 \sin x<0$
$\sin x \geq 1 $
$\sin x \leq\left(-\frac{1}{2}\right) $
$\Rightarrow \sin x=1 $
$x =2 m \pi+\frac{\pi}{2}, m \in I $ ......(i)
$x \in\left[\frac{7 \pi}{6}, \frac{11 \pi}{6}\right] \Rightarrow x \in\left[2 n \pi+\frac{7 \pi}{6}, 2 n \pi+\frac{11 \pi}{6}\right], n \in I $....(ii)
$\therefore \text { Union of (i) and (ii) is the domain of fog(x). } $