2022 ACJC P2 Q1

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

\(\mathrm{f}\left( x \right)=1-ax\), \(x\in \mathbb{R}\), where \(a\) is a real constant.

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

\(\mathrm{g}\left( x \right)=\left\{ \begin{matrix} {{x}^{2}}+3\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\le x\le 2 \\ 7-x\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2<x\le 7 \\ \end{matrix} \right.\)

1. Find the set of possible values of \(a\) such that \({{\mathrm{f}}^{-1}}\) exist.[1]
2. If \(a=2\), describe a sequence of transformations that transform the graph of \(y=\mathrm{g}\left( x \right)\) onto the graph of \(y=\mathrm{fg}\left( x \right)\).[3]
3. The function \(\mathrm{gg}\) is defined by
   \(\mathrm{gg}\left( x \right)\left\{ \begin{matrix} {{\mathrm{h}}_{1}}\left( x \right)\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,0\le x\le 2 \\ {{\mathrm{h}}_{2}}\left( x \right)\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,2<x<5 \\ {{\mathrm{h}}_{3}}\left( x \right)\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{5}\le x\le \mathrm{7} \\ \end{matrix} \right.\)
   Find \({{\mathrm{h}}_{1}}\left( x \right)\), \({{\mathrm{h}}_{2}}\left( x \right)\) and \({{\mathrm{h}}_{3}}\left( x \right)\).[3]