2020 YIJC P1 Q7

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

\(\begin{matrix} & \mathrm{f}:x\mapsto {{x}^{2}}+3x+1,x\in \mathbb{R},x\le -\frac{3}{2}, \\ & \mathrm{g}:x\mapsto 3+{{\mathrm{e}}^{-x}},x\in \mathbb{R}. \\ \end{matrix}\)

1. Show that \(\mathrm{f}\) has an inverse.[1]
2. Give a reason why the composite function \({{\mathrm{f}}^{-1}}\mathrm{g}\) exists. Find \({{\mathrm{f}}^{-1}}\mathrm{g}\left( x \right)\) and state the domain of \({{\mathrm{f}}^{-1}}\mathrm{g}\).[6]

The function \(\mathrm{h}\) is defined as follows.

\(\mathrm{h}:x\mapsto \ln \left( x-k+1 \right),x\in \mathbb{R},x\ge k,\) where \(k<0\).

3. Find the value of \(k\) such that the range of \(\mathrm{hg}\) is given by \(\left( \ln 5,\infty \right)\).[2]