Mathematics

Problem

Let

f:D\to\mathbb{R},\;f(x)=\sin x,\qquad g(x)=\arcsin x\;(-1\le x\le1).

(i) Determine all subsets

D\subseteq\mathbb{R}

for which the composition

(g\circ f)(x)

is well defined. (ii) Among these, identify the largest such subset

D_\text{max}

and write the domain of

(g\circ f)

when

D=D_\text{max}

. (iii) For

D=D_\text{max}

, examine whether

(f\circ g)(x)

is defined; if so, state its domain and find the composition.