Question

If R is a set of real numbers and $f:R\to R$ is given by the relation $f\left(x\right)=sinx,x\in R$ and mapping $g:R\to R$ by the relation $g\left(x\right)=x^2,x\in R$ then $fog=gof$.
If true enter 1 else 0

Answer 1

Given $f:R\to R$ as
$f\left(x\right)=\sin x$
And $g:R\to R$ as
$g\left(x\right)=x^2$
$\left(fog\right)\left(x\right)=f\left(g\right(x\left)\right)$
$=f\left(x^2\right)$
$=\sin x^2$
Now, $\left(gof\right)\left(x\right)=g\left(f\right(x\left)\right)$
$=g\left(\sin x\right)$
$={\sin }^{2}x$
So, $\left(fog\right)\left(x\right)\ne \left(gof\right)\left(x\right)$
$\Rightarrow fog\ne gof$