<p>A function \(\mathrm{h}\) is self-inverse if&nbsp;\(\mathrm{h}(x) = \mathrm{h}^{-1}(x)\) for all \(x\) in the domain of \(\mathrm{h}\).</p>
<p>The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by \[\mathrm{f}(x) = 2 + \frac{1}{x-2}, \quad \text{for}\hspace{0.5em} x \in \mathbb{R}, x &gt; 2\] \[\mathrm{g}(x) = \begin{cases} 3x - 1 & \text{for}\hspace{0.5em} x \le 1, \\ (x - 1)^2 + 2 & \text{for}\hspace{0.5em} x &gt; 1. \end{cases}\]</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{f}(x)\). With the aid of your graph, explain why \(\mathrm{f}\) has an inverse.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;\(\mathrm{f}\) is self-inverse and find \(\mathrm{f}^2(x)\).<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Hence, or otherwise, evaluate \(\mathrm{f}^{2025}(4)\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find an expression for&nbsp;\(\mathrm{g}\mathrm{f}\) and state its domain.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Find the range of \(\mathrm{g}\mathrm{f}\).<span class="inline-marks">[2]</span></div></li></ol>

A function \(\mathrm{h}\) is self-inverse if \(\mathrm{h}(x) = \mathrm{h}^{-1}(x)\) for all \(x\) in the domain of \(\mathrm{h}\).

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by \[\mathrm{f}(x) = 2 + \frac{1}{x-2}, \quad \text{for } x \in \mathbb{R}, x > 2\] \[\mathrm{g}(x) = \begin{cases} 3x - 1 & \text{for } x \le 1, \\ (x - 1)^2 + 2 & \text{for } x > 1. \end{cases}\]

Sketch the graph of \(y = \mathrm{f}(x)\). With the aid of your graph, explain why \(\mathrm{f}\) has an inverse.

Show that \(\mathrm{f}\) is self-inverse and find \(\mathrm{f}^2(x)\).

Hence, or otherwise, evaluate \(\mathrm{f}^{2025}(4)\).

Find an expression for \(\mathrm{g}\mathrm{f}\) and state its domain.

Find the range of \(\mathrm{g}\mathrm{f}\).

2024 CJC Promo Q11

2024 CJC Promo Q11