It is given that \(\mathrm{f}:x\mapsto(x-4)^2-|x-4|\), \(0\leq x\leq\lambda\), where \(\lambda\) is a constant.
Find the greatest value of \(\lambda\) for which the function \(\mathrm{f}^{-1}\) exists.[1]
The function \(\mathrm{g}(x)\) is defined by \[\mathrm{g}(x)=\begin{cases}x+2&\text{for}\hspace{0.5em}-2\leq x\leq0,\\-x-2&\text{for}\hspace{0.5em}0<x\leq13.\end{cases}\]
Sketch the graph of \(y=\mathrm{g}(x)\), stating the coordinates of the endpoints.[2]
Considering the result in part (a), show that the composite function \(\mathrm{gf}\) exists.[1]
Find the range of function \(\mathrm{gf}\).[1]
Without finding \(\mathrm{gf}\) and \(\mathrm{g}^{-1}\), find the exact value of \(x\) for which \(\mathrm{gf}(x)=-3\).[3]