$\text{ If }f\left(x\right)=2x+\left|x\right|,g\left(x\right)=\frac{1}{3}(2x-|x\left|\right)\text{ and }h\left(x\right)=f\left(g\right(x\left)\right)\text{ then domain of }{\sin }^{-1}\underset{n\text{ times }}{\underbrace{\left(h\right(h\left(h\right(h\dots h\left(x\right)\dots \left)\right)\left)\right)}}\text{ is }$

$$\begin{gathered} f\left(x \right)= 2x+x, x\ge 0 \\ =2x-x, x<0 \end{gathered}$$

 then $$\begin{gathered} f\left(x\right)=3x ,x\ge 0 \\ =x, x<0 \end{gathered}$$

 $$\begin{gathered} g\left(x\right)= \frac{1}{3}\left(2x-x\right), if x\ge 0 \\ =\frac{1}{3}\left(2x+x\right),x<0 \end{gathered}$$              $$\begin{gathered} \Rightarrow g\left(x\right)=\frac{x}{3 }, x\ge 0 \\ =x,x<0 \end{gathered}$$

$$\begin{gathered} therefore f\left(g\left(x\right)\right)=3\left(\frac{x}{3}\right) = x, if x\ge 0 \\ = x , if x<0 \end{gathered}$$         $\Rightarrow h\left(x\right) =x , for all x\in R$

 ${\sin }^{-1}\left(h\left(h\left(h......h\left(x\right).....\right)\right)\right)={\sin }^{-1}\left(x \right) , it\text{'}s domain is=[-1,1]$