Let f(x)=x^2 and g(x)= x for all x ∈ R . Then, the set of all x...

f(x)=x^2, g(x)=\sin x (g \circ f)(x)=\sin x^2$go(gof)$(x)=\sin $\left(\sin x^2\right)$(f \circ g \circ g \circ f)(x)=\left(\sin \left(\sin x^2\right)\right)^2$Again,$(g \circ f)(x)=\sin x^2$(gogof)$(x)=\sin $\left(\sin x^2\right)$ Given, $($ fogogof $)(x)=($ gogof $)(x)$ $\Rightarrow \quad$\left(\sin \left(\sin x^2\right)\right)^2=\sin \left(\sin x^2\right)$\Rightarrow \sin \left(\sin x^2\right)\left\{\sin \left(\sin x^2\right)-1\right\}=0$ $\Rightarrow \sin$\left(\sin x^2\right)=0$or$\sin $\left(\sin x^2\right)=1$ $\Rightarrow \sin x^2=0$ or $\sin x^2=\frac{\pi}{2}$ $\therefore \quad x^2=n \pi$ (i.e. not possible as $-1 \leq \sin \theta \leq 1$ )