N2024 P2 Q3

The function \(\mathrm{f}\) is such that \(\mathrm{f}(x) = -x^2 + 4x + 7\), for \(x \in \mathbb{R}\).

1. Find the range of \(\mathrm{f}\).[2]
2. Sketch the graph of \(\mathrm{f}\), giving the exact coordinates of the points where the curve crosses the axes.[2]

The function \(\mathrm{g}\) is such that \(\mathrm{g}(x) = \frac{2}{x} + 1\), for \(1 \leqslant x \leqslant 10\).

3. Explain how you know \(\mathrm{g}^{-1}\) exists.[1]
4. Find the value of \(\mathrm{f}\mathrm{g}^{-1}(1.5)\).[3]