2026 HCI Promo Q5

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

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

\[\mathrm{f}(x)=\frac{ax+b}{x-a},\quad\text{for}\hspace{0.5em}x\in\mathbb{R},\ x\ne a,\]

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

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

1. Show that \(\mathrm{f}^2(x)=x\).[1]
2. Hence, or otherwise, find a condition that \(a\) and \(b\) must satisfy.[2]
3. State the possible values of \(\mathrm{f}^n\left(-\dfrac{b}{a}\right)\) in terms of \(a\) and \(b\), where \(n\) is a positive integer.[2]

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

\[\mathrm{h}(x)=\begin{cases}\sqrt{10+3x},&\text{for}\hspace{0.5em}x\in\mathbb{R},\ x\geq0,\\x^2-4x+2,&\text{for}\hspace{0.5em}x\in\mathbb{R},\ x<0.\end{cases}\]

4. Given that \(a=-2\) and \(b=3\), find the exact value of \(\mathrm{hf}(1)\).[2]