2018 ACJC P1 Q5

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

Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by

\(\mathrm{f}:x\mapsto \frac{5x-3}{x-5}\), \(x\in \mathbb{R}\), \(x\ne a\), where \(a\) is a constant,

\(\mathrm{g}:x\mapsto \ln x\), \(x\in \mathbb{R}\), \(x\ge {{\mathrm{e}}^{10}}\).

1. State the value of \(a\) and explain why this value has to be excluded from the domain of \(\mathrm{f}\).[2]
2. Show that \(\mathrm{f}\) is self-inverse.[2]
3. Find the exact values of \(b\) such that \({{\mathrm{f}}^{4}}\left( b \right)-2={{\mathrm{f}}^{-1}}\left( b \right)\).[2]
4. Find the exact range of \(\mathrm{fg}\).[2]