<p>It is given that</p>
<p style="text-align: center">\[\mathrm{f}(x) = \begin{cases} \quad\;\;\; \frac{1}{2}x^2, & \text{for}\hspace{0.5em} 1 \le x &lt; 2, \\ \frac{1}{4}(-3x + 14), & \text{for}\hspace{0.5em} 2 \le x &lt; 4, \end{cases}\]</p>
<p>and that&nbsp;\(\mathrm{f}(x) = \mathrm{f}(x + 3)\) for all real values of \(x\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State the value of \(\mathrm{f}(0)\).<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{f}(x)\) for&nbsp;\(0 \le x \le 5\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The function \(\mathrm{g}\) is given by \(\mathrm{g}(x) = \frac{4}{3}|x - 2|\) for&nbsp;\(x \in \mathbb{R}\). By sketching the graph of \(y = \mathrm{g}(x)\) on the same diagram as in <strong>part (b)</strong>, solve the inequality \(\mathrm{f}(x) &gt; \mathrm{g}(x)\).<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The function \(\mathrm{h}\) is given by \(\mathrm{h}(x) = \sqrt{3} \sin(\pi x) + \cos(\pi x) + 1\) for&nbsp;\(0 \le x \le 2\). Explain why the composite function \(\mathrm{hf}\) exists and find its range in exact form.<span class="inline-marks">[4]</span></div></li></ol>

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

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

Sketch the graph of \(y = \mathrm{f}(x)\) for \(0 \le x \le 5\).

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 \(\mathrm{f}(x) > \mathrm{g}(x)\).

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.

2024 RI Promo Q10

2024 RI Promo Q10