2023 YIJC P1 Q8

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

\(\mathrm{f}:x\mapsto \frac{1}{{{x}^{2}}-4ax+5{{a}^{2}}}\), \(x\in \mathbb{R}\), \(x < k\),

where \(a\) and \(k\) are positive constants.

1. Find the largest possible value of \(k\), in terms of \(a\), such that \({{\mathrm{f}}^{-1}}\) exists.[2]

It is now given that \(k=a\) and \(a > 1\).

2. Show that the composite function \({{\mathrm{f}}^{2}}\) exists.[2]
3. Find \({{\mathrm{f}}^{-1}}\left( x \right)\) in terms of \(x\) and \(a\).[3]
4. On the same diagram, sketch the graphs of \(\mathrm{f}\) and \({{\mathrm{f}}^{-1}}\), giving the equations of any asymptotes.[4]