Question

If R is a set of real numbers and $f:R\to R$ is given by the relation $f\left(x\right)$ = sin x, $xϵR$ and mapping g : $R\to R$ by the relation $g\left(x\right)=x^2$, $xϵR$, then prove that $fog\ne gof$

Answer 1

$f:R⟶R$

$f\left(x\right)=\sin x$ $x$ $ϵ$ $R$

$g:R⟶R$

$g\left(x\right)=x^2$ $x$ $ϵ$ $R$

$fog=f\left(g\right(x\left)\right)=f\left(x^2\right)=\sin x^{2}gof=g\left(f\right(x\left)\right)=g\left(\sin x\right)={\left(\sin x\right)}^2\therefore fog\ne fof\left[Proved\right]$