2026 RI P2 Q3

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

\[\mathrm{f}:x\mapsto a-\mathrm{e}^{|x-a|},\quad x\in\mathbb{R},\]

where \(a\) is a positive constant.

It is given that the range of \(\mathrm{f}\) is \((-\infty,k]\), where \(k>0\).

1. Find the range of values of \(a\).[2]
2. Show that \(\mathrm{f}^{-1}\) does not exist.[2]
3. If the domain of \(\mathrm{f}\) is restricted to \((-\infty,m]\), state the largest value of \(m\) in terms of \(a\) for which \(\mathrm{f}^{-1}\) exists.[1]
4. Using your answer found in part (c), find \(\mathrm{f}^{-1}\) in similar form.[3]

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

\[\mathrm{g}:x\mapsto\ln\left(1-\frac{x}{a}\right),\quad x<a.\]

5. Show that the composite function \(\mathrm{gf}\) exists and find the range of \(\mathrm{gf}\) in terms of \(a\).[3]