Question:
Let $f$ and $g$ be two functions from $R$ into $R$, defined by $f(x)=|x|+x$ and $g(x)=|x|-x$ for all $x \in R .$ Find $f \circ g$ and $g \circ f$.
Solution:
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 $x<0$
$g(f(x))=0$
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