Functions - Modulus Functions

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

It is given that \(\mathrm{g}\) is a self-inverse function and is defined by \(\mathrm{g} : x \mapsto \frac{x+a}{3x+b}\), for \(x \in \mathbb{R},\, x > -\frac{b}{3}\),

where \(a\) and \(b\) are constants and \(\mathrm{g}(1) = 5\).

1. Show that \(a = 9\) and \(b = -1\).
2. Find \(\mathrm{g}^{2021}(1)\).

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

\[\mathrm{h} : x \mapsto \left\vert{} 1-x \right\vert{}(x+5),\text{ for}\hspace{0.5em} x \in \mathbb{R},\, x \ge 2.\]

3. Given that \(\mathrm{h}^{-1}\) exists, find \(\mathrm{h}^{-1}\) in similar form.
4. Show that \(\mathrm{gh}\) exists and find the exact range of \(\mathrm{gh}\).