2023 ASRJC P2 Q4

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

\(\mathrm{f}:x\mapsto \left| {{\mathrm{e}}^{x-3}}-1 \right|\), \(x\le 3\)

\(\mathrm{g}:x\mapsto {{a}^{2}}-{{x}^{2}}\), \(x\in \mathbb{R}\), \(x<0\), where \(0 < a < 1\).

1. Find \({{\mathrm{f}}^{-1}}\) in similar form.[3]
2. Sketch the graphs of \(y=\mathrm{g}\left( x \right)\), \(y={{\mathrm{g}}^{-1}}\left( x \right)\) and \(y={{\mathrm{g}}^{-1}}\mathrm{g}\left( x \right)\) on the same diagram.[3]
3. Find the exact solution of \(\mathrm{g}\left( x \right)={{\mathrm{g}}^{-1}}\left( x \right)\), leaving your answer in terms of \(a\).[2]
4. Show that the composite function \(\mathrm{fg}\) exists and find the range of \(\mathrm{fg}\), leaving your answer in terms of \(a\).[2]