Composite Functions

Let two functions are defined as $g\left(x\right)=\left\{\begin{array}{l}{x}^2, -1\le x<2\\ x+2, 2\le x\le 3\end{array}\right.$ and $f\left(x\right)=\left\{\begin{array}{l}x+1, x\le 1\\ 2x+1, 1<x\le 2\end{array}\right.$, then find $gof$.

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

$f\left(x\right)=\left\{\begin{array}{l}x-1, -1\le x\le 0\\ x^2, 0<x\le 1\end{array}\right.$ and $g\left(x\right)=\sin x$, then find $h\left(x\right)=f\left(\left|g\left(x\right)\right|\right)+\left|f\left(g\left(x\right)\right)\right|$.

Let $f\left(x\right)=x^2-2x, x\in R$ and $g\left(x\right)=f\left(f\left(x\right)-1\right)+f\left(5-f\left(x\right)\right),$ show that $g\left(x\right)\ge 0,\forall x\in R$.