The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x \mapsto \frac{1}{2}+e^{-x}\), \(x \in \mathbb{R}\), \(x \ge 0\).
Sketch the graph of \(y=\mathrm{f}(x)\), \(y=\mathrm{f}^{-1}(x)\) and \(y=\mathrm{f}^{-1}\mathrm{f}(x)\) on the same diagram, showing clearly their relationship.[3]
The function \(\mathrm{g}\) is defined by \(\mathrm{g}:x \mapsto 1+4x-x^2\), \(x \le 2\).
Explain why \(\mathrm{g}^{-1}\) exists.[1]
Find \(\mathrm{g}^{-1}(x)\) and state its domain.[3]
Explain why the composite function \(\mathrm{gf}\) exists.[2]
Find the exact range of \(\mathrm{gf}\).[2]