<p>The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto 1 + \frac{3}{\mathrm{e}^x - 2}\) for \(x \in \mathbb{R}\), \(x &gt; \ln 2\).</p>
<ol class="abc-list"><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>Another function \(\mathrm{g}\) is defined by</p>
<p>\[\mathrm{g} : x \mapsto \begin{cases} 2(x - 1)^2 + 2 & \text{for } x \le 0, \\ 2 + |2 - x| & \text{for } 0 &lt; x \le 3. \end{cases}\]</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">Sketch the graph of \(y = \mathrm{g}(x)\) and explain why the composite function \(\mathrm{f}^{-1}\mathrm{g}\) exists.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the exact value of \(k\) such that \(\mathrm{f}^{-1}\mathrm{g}(k) = \ln \frac{5}{2}\).</span><span class="inline-marks">[3]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto 1 + \frac{3}{\mathrm{e}^x - 2}\) for \(x \in \mathbb{R}\), \(x > \ln 2\).

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

Another function \(\mathrm{g}\) is defined by

\[\mathrm{g} : x \mapsto \begin{cases} 2(x - 1)^2 + 2 & \text{for } x \le 0, \\ 2 + |2 - x| & \text{for } 0 < x \le 3. \end{cases}\]

Sketch the graph of \(y = \mathrm{g}(x)\) and explain why the composite function \(\mathrm{f}^{-1}\mathrm{g}\) exists.

Hence find the exact value of \(k\) such that \(\mathrm{f}^{-1}\mathrm{g}(k) = \ln \frac{5}{2}\).

2025 RI P1 Q7

2025 RI P1 Q7