2024 ASRJC P1 Q8

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

\(\mathrm{f}:x\mapsto \left| 4+2x-{{x}^{2}} \right|\), \(x\in \mathbb{R}\), \(x\ge 3.5\),

\(\mathrm{g}:x\mapsto 4+\mathrm{e}{{\,}^{ax}}\), \(x\in \mathbb{R}\), \(x\ge -1\),

where \(a > 0\).

1. Find \({{\mathrm{f}}^{-1}}\left( x \right)\) and state its domain.[3]
2. Find the value of \(x\) for which \({{\mathrm{f}}^{-1}}\left( x \right)=\mathrm{f}\left( x \right)\).[2]
3. Show that the composite function \(\mathrm{fg}\) exists and express the exact range of \(\mathrm{fg}\) in the form of \(A+B{{\mathrm{e}}^{-a}}+C{{\mathrm{e}}^{-2a}}\), where \(A\), \(B\) and \(C\) are real constants.[4]
4. Without the use of a graphing calculator, solve the inequality \(\frac{\mathrm{g}\left( x \right)}{{{x}^{2}}-2x-2}\ge 0\).
   Leave your answer in exact form.[3]