2022 ASRJC J2 MYCT P1 Q8

The function \(\mathrm{f}\) is defined by

\(\mathrm{f}:x\mapsto 1+ax-{{x}^{2}}\), \(x\in \mathbb{R}\), \(x\le k\)

where \(a\) is a positive constant.

1. Explain with a sketch why the largest value of \(k\) for which \({{\mathrm{f}}^{-1}}\) exists is \(\frac{a}{2}\).
   With this value of \(k\), find \({{\mathrm{f}}^{-1}}\) in a similar form.[5]
2. If \(k=1\) and \(a=2\), find the set of values of \(x\) for \(\mathrm{f }{{\mathrm{f}}^{-1}}\left( x \right)={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)\).[1]
3. The function \(\mathrm{g}\) is defined by
   \(\mathrm{g}:x\mapsto \left\{ \begin{matrix} 2+2x-{{x}^{2}},0\le x\le 1, \\ 1\,\,\,\,\,,1<x<2. \\ \end{matrix} \right.\)
   It is also given that \(\mathrm{g}\left( x \right)=\mathrm{g}\left( x+2 \right)\) for all real values of \(x\).
   Sketch \(y=\mathrm{g}\left( x \right)\) for \(0\le x\le 5\).[2]