2017 AJC MYCT Q7

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined as follows

\(\mathrm{f}:x\mapsto \frac{3x-a}{x-b}\), \(x\in \mathbb{R}\), \(x\ne b\),

\(\mathrm{g}:x\mapsto 3+8x-2{{x}^{2}}\), \(x>-1\),

where \(a\) and \(b\) are positive constants and \(a < b\).

1. By sketching the graph of \(y=\mathrm{f}(x)\), explain why \({{\mathrm{f}}^{-1}}\) exists and find \({{\mathrm{f}}^{-1}}(x)\).[4]
2. Find the set of values of \(b\) such that the composite function \(\mathrm{fg}\) exists. Hence find the range of \(\mathrm{fg}\) in terms of \(a\) and \(b\).[4]
3. If \({{\mathrm{f}}^{-1}}(x)=\mathrm{f}(x)\) for all values of \(x\), find the value of \(b\). Hence find the value of \({{\mathrm{f}}^{(2017)}}(a)\), leaving your answer in terms of \(a\).[3]