Let f and g be two functions from R into R

To Find: Inverse of f o g and g o f

Given: $f(x)=|x|+x$ and $g(x)=|x|-x$ for all $x \in R$

fog $(x)=f(g(x))=|g(x)|+g(x)=|| x|-x|+|x|-x$

Case 1$)$ when $x \geq 0$

$f(g(x))=0$ (i.e. $|x|-x)$

Case 2$)$ when $x<0$

$f(g(x))=-4 x$

$g \circ f(x)=g(f(x))=|f(x)|-f(x)=|| x|+x|-|x|-x$

Case 1 ) when $x \geq 0$

$g(f(x))=0($ i.e. $|x|-x)$

Case 2$)$ when $x<0$

$g(f(x))=0$