2024 ACJC P2 Q5

The function \(h\) is defined by \(h : x \mapsto \left[\ln(x+2)\right]^2 + 1\), \(x \in \mathbb{R}\), \(x > -2\).

1. The function \(\mathrm{h}^{-1}\) exists if the domain of \(\mathrm{h}\) is restricted to \(-2 < x \le k\). State the greatest possible value of \(k\).[1]

The function \(\mathrm{f}\) is defined by \[{\mathrm{f}(x) = \begin{cases} \left[\ln\left(x+2\right)\right]^2 + 1, & \text{for}\hspace{0.5em} x \in \mathbb{R}, -2 < x \leq -1, \\ \qquad \frac{1}{x+2}, & \text{for}\hspace{0.5em} x \in \mathbb{R}, x > -1. \end{cases}}\]

2. Sketch the graph of \(y = \mathrm{f}(x)\).[1]
3. Given that \(\mathrm{f}^{-1}\) exists, find \(\mathrm{f}^{-1}\) in a form similar to \(\mathrm{f}\).[4]
4. Show that \(\mathrm{f}^2\) exists and find its range.[2]
5. If \(\mathrm{f}^2(2) = \mathrm{f}(x)\), find \(x\).[2]