2024 RI Promo Q10

It is given that

\[\mathrm{f}(x) = \begin{cases} \quad\;\;\; \frac{1}{2}x^2, & \text{for}\hspace{0.5em} 1 \le x < 2, \\ \frac{1}{4}(-3x + 14), & \text{for}\hspace{0.5em} 2 \le x < 4, \end{cases}\]

and that \(\mathrm{f}(x) = \mathrm{f}(x + 3)\) for all real values of \(x\).

1. State the value of \(\mathrm{f}(0)\).[1]
2. Sketch the graph of \(y = \mathrm{f}(x)\) for \(0 \le x \le 5\).[3]
3. The function \(\mathrm{g}\) is given by \(\mathrm{g}(x) = \frac{4}{3}|x - 2|\) for \(x \in \mathbb{R}\). By sketching the graph of \(y = \mathrm{g}(x)\) on the same diagram as in part (b), solve the inequality \(\mathrm{f}(x) > \mathrm{g}(x)\).[4]
4. The function \(\mathrm{h}\) is given by \(\mathrm{h}(x) = \sqrt{3} \sin(\pi x) + \cos(\pi x) + 1\) for \(0 \le x \le 2\). Explain why the composite function \(\mathrm{hf}\) exists and find its range in exact form.[4]