2025 NJC P1 Q11

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

\[\mathrm{f}(x) = \begin{cases} \frac{a}{2} + \frac{4}{3}\left(x - \frac{1}{2}\right)^2 & \text{for}\hspace{0.5em} x \in \mathbb{R}, \frac{1}{2} < x < 2, \\ \frac{a}{x} & \text{for}\hspace{0.5em} x \in \mathbb{R}, x \ge 2, \end{cases}\]

where \(a\) is a positive constant.

1. Find the range of \(\mathrm{f}\).[3]
2. Find \(\mathrm{f}^{-1}(x)\) and state its domain.[3]

The function \(\mathrm{g}\) is defined by \(\mathrm{g}(x) = 3 + \mathrm{e}^x\), for \(x \in \mathbb{R}\).

3. Show that \(\mathrm{f}\mathrm{g}\) exists.[1]
4. Find the exact value of \(k\) for which \(\mathrm{f}\mathrm{g}(k) = \frac{a}{7}\).[3]