2016 HCI Promo Q6

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

\(\mathrm{f}:x\mapsto \left\{ \begin{matrix} 4{{a}^{2}}-{{(x-a)}^{2}}\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}~\,\,\,\,\,\,0\le x\le 2a, \\ 5{{a}^{2}}-ax\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{for}~\,\,\,2a<x\le 4a, \\ \end{matrix} \right.\)

\(\mathrm{g}:x\mapsto x+a\), \(x\in \mathbb{R}\), \(-a<x\le 2a\),

where \(a>1\).

1. Sketch the graph of \(y=\mathrm{f}\left( x \right)\) for \(0\le x\le 4a\). If the maximal domain of \(\mathrm{f}\) is restricted to \(\mathrm{D}=\left[ \alpha ,\beta \right]\) such that the inverse function \({{\mathrm{f}}^{-1}}\) exists and the range of \(\mathrm{f}\) remains unchanged, find the maximal domain \(\mathrm{D}\).[3]
2. For the maximal domain \(\mathrm{D}\) found in part (i), determine the value of \(t\), in terms of \(a\), such that \({{\mathrm{f}}^{-1}}\left( t \right)=\frac{5}{2}a\).[2]
3. Show that the composite function \(\mathrm{fg}\) exists and find the range of \(\mathrm{fg}\), expressing your answer in terms of \(a\).[3]