2024 CJC P2 Q2

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

\(\mathrm{f}:x\mapsto \frac{3-2x}{x+2}\), for \(x\in \mathbb{R}\), \(x>-2\),

\(\mathrm{g}:x\mapsto \left| x \right|-1\), for \(x\in \mathbb{R}\).

1. Find \({{\mathrm{f}}^{-1}}\left( x \right)\) and state its domain.[3]
2. Hence find \({{\mathrm{f}}^{2024}}\left( 1 \right)\).[1]
3. Show that \(\mathrm{fg}\) exists and find its range.[3]
4. Solve \(\mathrm{f}\left( x \right)>\mathrm{g}\left( x \right)\), giving your answer in exact form.[4]