<p>The function \(\mathrm{f}\) is defined by</p>
<p>\[\mathrm{f}(x) = \begin{cases} \frac{a}{2} + \frac{4}{3}\left(x - \frac{1}{2}\right)^2 & \text{for } x \in \mathbb{R}, \frac{1}{2} &lt; x &lt; 2, \\ \frac{a}{x} & \text{for } x \in \mathbb{R}, x \ge 2, \end{cases}\]</p>
<p>where \(a\) is a positive constant.</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the range of \(\mathrm{f}\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find \(\mathrm{f}^{-1}(x)\) and state its domain.</span><span class="inline-marks">[3]</span></div></li></ol>
<p>The function \(\mathrm{g}\) is defined by \(\mathrm{g}(x) = 3 + \mathrm{e}^x\), for \(x \in \mathbb{R}\).</p>
<ol class="i-list" start="3"><li><div class="list-item-content"><span class="list-item-text">Show that \(\mathrm{f}\mathrm{g}\) exists.</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the exact value of \(k\) for which \(\mathrm{f}\mathrm{g}(k) = \frac{a}{7}\).</span><span class="inline-marks">[3]</span></div></li></ol>

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

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

Find \(\mathrm{f}^{-1}(x)\) and state its domain.

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

Show that \(\mathrm{f}\mathrm{g}\) exists.

Find the exact value of \(k\) for which \(\mathrm{f}\mathrm{g}(k) = \frac{a}{7}\).

2025 NJC P1 Q11

2025 NJC P1 Q11