2025 HCI Promo Q8

The function \(\mathrm{f}\) and \(\mathrm{g}\) are defined as

\(\mathrm{f}: x \mapsto \pi x, \quad x \in \mathbb{R}.\)
\(\mathrm{g}: x \mapsto \begin{cases} \quad \sin x & \text{for}\hspace{0.5em} 0 \leq x \leq \frac{\pi}{2}, \\ \cos x + 1 & \text{for}\hspace{0.5em} \frac{\pi}{2} < x \leq \pi. \end{cases}\)

1. Find \(\mathrm{f}^{25}(x)\).[1]
2. Sketch the graph of \(y = \mathrm{g}(x)\) for \(0 \le x \le \pi\).[2]
3. If the domain of \(\mathrm{g}\) is restricted to \(D = (a, \pi]\) such that \(\mathrm{g}^{-1}\) exists, state the smallest exact value of \(a\). Hence find \(\mathrm{g}^{-1}(x)\) and state the domain of \(\mathrm{g}^{-1}\).[3]

4. For the domain \(D\) found in part (c), show that the composite function \(\mathrm{fg}\) exists and find the exact range of \(\mathrm{fg}\).

   [3]
5. Without finding \(\mathrm{f}^{-1}\) and \(\mathrm{fg}\), find the exact solution of \(\mathrm{f}^{-1}(x) = \mathrm{g}\left(\frac{3\pi}{4}\right)\).[3]