2019 DHS Promo Q8

The function \(\mathrm{f}\) is defined by

\(\mathrm{f}:x\mapsto \frac{2x}{x-2}\), for \(x\in \mathbb{R},x\ne 2\).

1. Find \({{\mathrm{f}}^{-1}}\left( x \right)\), stating the domain of \({{\mathrm{f}}^{-1}}\).[2]
2. Solve the equation \(\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)\).[1]

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

\(\mathrm{g}:x\mapsto \frac{2}{x-3}\), for \(x\in \mathbb{R},x\ne 3,x\ne 4\).

3. Find \(\mathrm{fg}\left( x \right)\).[1]
4. Solve the inequality \(\mathrm{fg}\left( x \right) < x\) for all \(x\) in the domain of \(\mathrm{fg}\).[3]
5. Find the range of \(\mathrm{fg}\).[2]