Question

Let $R$ be the set of real numbers. If $f:R\to R;f\left(x\right)=2x^{1/2}$ and $g:R\to R;g\left(x\right)=4x^2$.
Then find $fog\left(x\right)$ and $gof\left(x\right)$. Also find the relation between $fog\left(x\right)$ and $gof\left(x\right)$.

Answer 1

Given, $f:R\to R;f\left(x\right)=2x^{1/2}$ and
$g:R\to R;g\left(x\right)=4x^2$
$fog\left(x\right)=f\left(g\right(x\left)\right)$
$=f\left(4x^2\right)$
$=2(4x^2{)}^{1/2}$
$=2\left(\right(2x{)}^2{)}^{1/2}$
$=4x$

$gof\left(x\right)=g\left(f\right(x\left)\right)$
$=g\left(2x^{1/2}\right)$
$=4(2x^{1/2}{)}^2$
$=16x$

So, $gof\left(x\right)=4fog\left(x\right)$