Question

Find $gof$ and $fog,$if $f:R\to R$ and $g:R\to R$ are given by $f\left(x\right)=\cos x$ and $g\left(x\right)=3x^2$. Show that $gof\ne fog$

Answer 1

Given: $f\left(x\right)=\cos x$ and $g\left(x\right)=3x^2$
$f\left(x\right)=\cos x$
$\Rightarrow f\left(g\right(x\left)\right)=\cos g\left(x\right)$
$\Rightarrow fog\left(x\right)=\cos \left(3x^2\right)...\left(1\right)$

$g\left(x\right)=3x^2$
$\Rightarrow g\left(f\right(x\left)\right)=3f(x{)}^2$
$\Rightarrow gof\left(x\right)=3{\cos }^{2}x...\left(2\right)$
From (1) and (2),
$gof\ne fog$.