2016 JJC P1 Q6

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 \(\mathrm{f}\).

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

\(\mathrm{g}:x\mapsto \sqrt{\frac{{{x}^{2}}+2}{{{x}^{2}}-1}},\mathrm{ }x>1\) .

1. Sketch the curve \(y=\mathrm{g}\left( x \right)\) stating the equations of the asymptotes clearly.[2]
2. Define \({{\mathrm{g}}^{-1}}\) in a similar form and show that \(\mathrm{g}\) is self-inverse.[4]
3. Show that \({{\mathrm{g}}^{2}}(x)=x\) and that \({{\mathrm{g}}^{3}}(x)=\mathrm{g}(x)\) . Hence find the values of \(x\) for which \(4-{{\mathrm{g}}^{50}}(x)={{\left[ {{\mathrm{g}}^{51}}(x) \right]}^{2}}\).[4]