2022 HCI P1 Q6

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

\(\mathrm{f}:x\mapsto \left\{ \begin{matrix} 5+\ln x\,\,\,\,\,\,\,\mathrm{for}\,\,0<x<1, \\ -{{x}^{2}}+6x\,\,\,\,\mathrm{for }1\le x\le a, \\ \end{matrix} \right.\)

where \(a\) is a constant.

1. Given that \(\mathrm{f}\) has an inverse, determine the range of values of \(a\).[2]
2. Given that \(a\) satisfies the range of values found in part (i), sketch the graph of \(\mathrm{f}\), indicating clearly the coordinates of the end point(s) in terms of \(a\).[2]
3. Find \({{\mathrm{f}}^{-1}}\) in similar form.[3]