N2009 P2 Q3

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

\[\mathrm{f} : x \mapsto \frac{ax}{bx - a}, \quad \text{for}\hspace{0.5em} x \in \mathbb{R}, \ x \neq \frac{a}{b},\]

where \(a\) and \(b\) are non-zero constants.

1. Find \(\mathrm{f}^{-1}(x)\). Hence or otherwise find \(\mathrm{f}^2(x)\) and state the range of \(\mathrm{f}^2\).[5]
2. The function \(\mathrm{g}\) is defined by \(\mathrm{g} : x \mapsto \frac{1}{x}\) for all real non-zero \(x\). State whether the composite function \(\mathrm{fg}\) exists, justifying your answer.[2]
3. Solve the equation \(\mathrm{f}^{-1}(x) = x\).[3]