$\text{ Let }f\left(x\right)=\sin x\text{ and }g\left(x\right)={\log }_e\left|x\right|\text{ . If the ranges of the composition functions fog and gof are }{R}_1\text{ and }{R}_2\text{ , respectively, then }$

$f\left(x\right) = \sin x and g\left(x\right)=\log \left|x\right|$

$\Rightarrow fog\left(x\right) =f\left(g\left(x\right)\right)=f\left(\log \left|x\right|\right) =\sin \left(\log \left|x\right|\right) \in \left[-1,1\right] \left( \sin ce \log \left|x\right|, range is R\right)$

$R_1=\left\{u/ -1\le u\le 1\right\}$ 

$also , gof\left(x\right) =g\left(f\left(x\right)\right)=g\left(\sin x\right)=\log \left|\sin x\right|$

     $\sin ce 0\le \left|\sin x\right|\le 1 \Rightarrow -\infty <\log \left|\sin x\right|\le 0$    

$R_2=\left\{v/-\infty <v\le 0\right\}$