2009 TJC P1 Q8

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by

\(\begin{matrix} & \mathrm{f}:x\mapsto -{{(x-2)}^{2}}+4,x\in \mathbb{R},x\le 2, \\ & \mathrm{g}:x\mapsto \frac{1}{{{(x-7)}^{2}}-4},x\in \mathbb{R},x\le k,x\ne 5,x\ne 9 \\ \end{matrix}\)

1. 1. Sketch the graphs of \(\mathrm{f}\) and \({{\mathrm{f}}^{-1}}\) on the same axes, showing the relationship between the two graphs.
   2. If \(\mathrm{f}(\alpha )={{\mathrm{f}}^{-1}}(\alpha )\), show that \({{\alpha }^{2}}-3\alpha =0\).
2. State the maximum value of \(k\)such that the inverse of \(\mathrm{g}\) exists. With this value of \(k\), find
   1. the inverse of \(\mathrm{g}\), stating its domain, and
   2. the range of \(\mathrm{gf}\).