A) \[\left\{ 2n\pi +\frac{\pi }{2} \right\},n\in I\]
B) \[\underset{n\in I}{\mathop{\bigcup }}\,\]\[\left( 2n\pi +\frac{7\pi }{6},2n\pi +\frac{11\pi }{6} \right)\]
C) \[\left( 2n\pi +\frac{7\pi }{6} \right),n\in I\]
D) \[\{(4m+1)\frac{\pi }{2}:m\in I\}\underset{n\in I}{\mathop{\bigcup }}\,\left[ \left( 2n\pi +\frac{7\pi }{6},2n\pi +\frac{11\pi }{6} \right) \right]\]
Correct Answer: D
Solution :
[d] For \[(fog)(x)\]to exists range of g\[\subseteq \]domain of f. \[\therefore \] Domain of \[f\Rightarrow 3\left| x \right|-x-2\ge 0\] \[\Rightarrow 3\left| x \right|-x\ge 2\] When \[x\ge 0\Rightarrow x\ge 1\]when \[x<0\Rightarrow x<-\frac{1}{2}\] \[\therefore \,\,\,\,\sin x\ge 1\,\,and\,\,x<-\frac{1}{2}\] for \[f\{g(x)\}\] to exists. i.e. \[\sin \,\,x=1\,\,and\,\,-1\,\le \,\sin \,x<-\frac{1}{2}\] \[\therefore x=(4m+1)\frac{\pi }{2}\] and \[2n\pi +\frac{7\pi }{6}\le x\le 2n\pi +\frac{11\pi }{6}\] \[ie,\] \[\left\{ (4m+1)\frac{\pi }{2}:m\in I \right\}\underset{n\in 1}{\mathop{\bigcup }}\,\left\{ 2n\pi +\frac{7\pi }{6}\le 2n\pi +\frac{11\pi }{6} \right\}\]
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