Question

Find g o f and f o g 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 g o f $\ne$ f o g.

Answer 1

Given that $f\left(x\right)=cosx$ and $g\left(x\right)=3x^2$

Given $f:R\to R$ and $g:R\to R$,

$\therefore fog:R\to R$ and $gof:R\to R$

$fog\left(x\right)=f\left(g\right(x\left)\right)=f\left(3x^2\right)=cos\left(3x^2\right)$

$gof\left(x\right)=g\left(f\right(x\left)\right)=g\left(cosx\right)=3co{s}^{2}x$

Since $cos\left(3x^2\right)\ne 3co{s}^{2}x$.

$\therefore gof\ne fog$