2008 HCI P1 Q8

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

\(\begin{matrix} & \mathrm{f}:x\mapsto -{{e}^{\left| 3-x \right|}},x\in \mathbb{R}, \\ & \mathrm{g}:x\mapsto a-{{(x-1)}^{2}},x\in \mathbb{R}\mathrm{ and }a\mathrm{ is a constant} \\ \end{matrix}\)

Sketch the graph of \(y=\mathrm{f}(x)\), and show that \(\mathrm{f}\) does not have an inverse.

1. The function \(\mathrm{f}\) has an inverse if its domain is restricted to \(x\ge b\). State the smallest possible value of \(b\) and define, in similar form, the inverse function \({{\mathrm{f}}^{-1}}\)corresponding to this domain for \(\mathrm{f}\).
2. Find the largest value of \(a\) such that the composite function \({{\mathrm{f}}^{-1}}\mathrm{g}\) exists and state the range of \({{\mathrm{f}}^{-1}}\mathrm{g}\) for this value of \(a\).