Question

Let $f,g:R\to R$ be two functions defined as $f\left(x\right)=\left|x\right|+x,g\left(x\right)=\left|x\right|-x,\forall x\in R$. Then, find $f\circ g$ and $g\circ f$.

Answer 1

$f\left(x\right)=|x|+x$

$g\left(x\right)=|x|-x$

$fog\left(x\right)=f\left(g\right(x\left)\right)=f(|x|-x)$

$⟹fog\left(x\right)=||x|-x|+|x|-x$

And--

$gof\left(x\right)=g\left(f\right(x\left)\right)=g(|x|+x)$

$⟹gof\left(x\right)=||x|+x|-|x|-x$