2024 SAJC P2 Q3

A function \(g\) is said to be self-inverse if \({{\mathrm{g}}^{-1}}\left( x \right)=\mathrm{g}\left( x \right)\) for all \(x\) in the domain of \(g\).

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

\(\mathrm{f}\left( x \right)=\frac{ax+b}{x-2}\), \(x\in \mathbb{R}\), \(x\ne 2\),

where \(a\) and \(b\) are real constants.

It is given that \(\mathrm{f}\) is self-inverse.

1. Find \(a\) and the value(s) that \(b\) cannot take.[4]
2. Explain why the composite function \({{\mathrm{f}}^{n}}\) always exists for all integers \(n > 1\).[1]
3. Hence find \({{\mathrm{f}}^{2025}}\left( 1 \right)\) in terms of \(b\).[2]

The function \(\mathrm{h}\) is defined for all \(x\in \mathbb{R}\) by

\(\mathrm{h}\left( x \right)=\left\{ \begin{matrix} 3x+2\,\,\,\,\,\,\,\,x\ge 0, \\ 2{{x}^{2}}\,\,\,\,\,\,\,\,\,\,\,\,\,x<0. \\ \end{matrix} \right.\)

4. Sketch the graph of \(\mathrm{h}\).[2]
5. Using the value of \(a\) found in (i) and given that \(b=1\), find \(\mathrm{hf}\left( \frac{1}{2} \right)\).[1]