<p>The function \(h\) is defined by \(h : x \mapsto \left[\ln(x+2)\right]^2 + 1\), \(x \in \mathbb{R}\), \(x &gt; -2\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">The function \(\mathrm{h}^{-1}\) exists if the domain of \(\mathrm{h}\) is restricted to \(-2 &lt; x \le k\). State the greatest possible value of \(k\).<span class="inline-marks">[1]</span></div></li></ol>
<p>The function \(\mathrm{f}\) is defined by \[{\mathrm{f}(x) = \begin{cases} \left[\ln\left(x+2\right)\right]^2 + 1, & \text{for}\hspace{0.5em} x \in \mathbb{R}, -2 &lt; x \leq -1, \\ \qquad \frac{1}{x+2}, & \text{for}\hspace{0.5em} x \in \mathbb{R}, x &gt; -1. \end{cases}}\]</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{f}(x)\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;\(\mathrm{f}^{-1}\) exists, find \(\mathrm{f}^{-1}\) in a form similar to \(\mathrm{f}\).<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">Show that&nbsp;\(\mathrm{f}^2\) exists and find its range.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-e" data-list-item-path="item-e"><div class="list-item-paragraph">If&nbsp;\(\mathrm{f}^2(2) = \mathrm{f}(x)\), find \(x\).<span class="inline-marks">[2]</span></div></li></ol>

The function \(h\) is defined by \(h : x \mapsto \left[\ln(x+2)\right]^2 + 1\), \(x \in \mathbb{R}\), \(x > -2\).

The function \(\mathrm{h}^{-1}\) exists if the domain of \(\mathrm{h}\) is restricted to \(-2 < x \le k\). State the greatest possible value of \(k\).

The function \(\mathrm{f}\) is defined by \[{\mathrm{f}(x) = \begin{cases} \left[\ln\left(x+2\right)\right]^2 + 1, & \text{for } x \in \mathbb{R}, -2 < x \leq -1, \\ \qquad \frac{1}{x+2}, & \text{for } x \in \mathbb{R}, x > -1. \end{cases}}\]

Sketch the graph of \(y = \mathrm{f}(x)\).

Given that \(\mathrm{f}^{-1}\) exists, find \(\mathrm{f}^{-1}\) in a form similar to \(\mathrm{f}\).

Show that \(\mathrm{f}^2\) exists and find its range.

If \(\mathrm{f}^2(2) = \mathrm{f}(x)\), find \(x\).

Functions

International Baccalaureate Higher Level Mathematics

2024 ACJC P2 Q5

2024 ACJC P2 Q5