If $f, \mathrm{~g}: \mathrm{R} \rightarrow \mathrm{R}$ be two functions defined as $f(x)=|x|+x$ and $\mathrm{g}(x)=|x|-x, \forall x \in \mathrm{R}$. Then find fog and gof. Hence find fog(-3), fog(5) and gof $(-2)$.
Given: $f(x)=|x|+x$
and $g(x)=|x|-x, \forall x \in \mathrm{R}$
$f o g=f(g(x))=|g(x)|+g(x)$
$=|| x|-x|+(|x|-x)$
Therefore,
$f(g(x))= \begin{cases}0 & x \geq 0 \\ 4 x & x<0\end{cases}$
$f o g= \begin{cases}4 x & x<0 \\ 0 & x \geq 0\end{cases}$
$g o f=g(f(x))=|f(x)|-f(x)$
$=|| x|+x|-(|x|+x)$
$g(f(x))= \begin{cases}0 & x \geq 0 \\ 0 & x<0\end{cases}$
Therefore, $g(f(x))=g o f=0$
Now, $f o g(-3)=(4)(-3)=-12 \quad$ (since, $f \circ g=4 x$ for $x<0)$
$f o g(5)=0 \quad$ (since, $f o g=0$ for $x \geq 0)$
$g o f(-2)=0 \quad$ (since, $g \circ f=0$ for $x<0)$