Question

Domain of given :$y=\sqrt{\log \left\{\sin \left({\frac{x}{3}}^2-1\right)\right\}}$

Answer 1

$y=\sqrt{{\log }_e\left\{\sin (\frac{x^2}{3}-1)\right\}}$

$y$ is defined if

${\log }_e\left\{\sin (\frac{x^2}{3}-1)\right\}\ge 0$

${\log }_e\left\{\sin (\frac{x^2}{3}-1)\right\}\ge {\log }_{e}1$

$\sin (\frac{x^2}{3}-1)\ge 1$

$\frac{x^2}{3}-1\ge {\sin }^{-1}1$

$\frac{x^2}{3}\ge \frac{\pi }{2}$

$x^2\ge \frac{3\pi }{2}+9$

$|x|\ge \sqrt{3\frac{\pi }{2}+9}$

$x\le -\sqrt{3\frac{\pi }{2}+9}$ OR $x\ge \sqrt{3\frac{\pi }{2}+9}$

$\therefore$ domain $xϵ(-\infty ,-\sqrt{\frac{3\pi }{2}+9}]\cup [\sqrt{\frac{3\pi }{2}+9},\infty )$