Question

If f(x) be defined on [−2, 2] and is given by $f\left(x\right)=\left\{\begin{array}{ll}-1,& -2\le x\le 0\\ x-1,& 0<x\le 2\end{array}\right.$and g(x) $=f\left(\left|x\right|\right)+\left|f\left(x\right)\right|$, find g(x).

Answer 1

Given:
$f\left(x\right)=\left\{\begin{array}{ll}-1,& -2⩽x⩽0\\ x-1,& 0<x⩽2\end{array}\right.$
Thus,
$g\left(x\right)=f\left(\left|x\right|\right)+\left|f\left(x\right)\right|$
$$\begin{gathered} =\left\{\begin{array}{ll}x-1+1,& -2⩽x⩽0\\ x-1+(-x+1),& 0<x<1\\ x-1+x-1,& 1\le x\le 2\end{array}\right. \\ =\left\{\begin{array}{ll}x,& -2⩽x⩽0\\ 0,& 0<x<1\\ 2x-2,& 1\le x\le 2\end{array}\right. \end{gathered}$$