2021 ACJC P1 Q3

The function \(\mathrm{f}\) is defined as \(\mathrm{f}:x\mapsto \frac{a}{2-x}\), \(x\in \mathbb{R}\), \(x\ne 0,2\).

1. Find \({{\mathrm{f}}^{2}}\) and \({{\mathrm{f}}^{-1}}\) in terms of \(a\), stating their domains clearly.[5]
2. Find the value of \(a\) such that \({{\mathrm{f}}^{2}}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)\) for all \(x\in \mathbb{R}\), \(x\ne 0,2,\frac{a}{2}\).[2]
3. Using the value of \(a\) found in (ii), find \({{\mathrm{f}}^{2021}}\left( x \right)\).[2]