<ol class="abc-list"><li><div class="list-item-paragraph" style="text-align: center">\[\mathrm{f} : x \mapsto \frac{\sqrt{2}(x-2)}{x^2+x-2}, \quad x \in \mathbb{R}, x &gt; 1,\]</div><div class="list-item-paragraph" style="text-align: center">\[\mathrm{g} : x \mapsto \mathrm{e}^{-x}-1, \quad x \in \mathbb{R}.\]</div><ol class="i-list"><li><div class="list-item-paragraph">Determine if the composite function \(\mathrm{fg}\) exists.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the exact range of \(\mathrm{gf}\).<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">The domain of \(\mathrm{f}\) is now restricted to \(0 &lt; x &lt; 1\). Find the value of \(\mathrm{g}^{-1}\mathrm{f}^{-1}(2)\).<span class="inline-marks">[3]</span></div></li></ol></li><li><div class="list-item-paragraph" style="text-align: center">\[\mathrm{h} : x \mapsto \sqrt{3}(\sin x + \cos x), \quad x \in \mathbb{R}, -\frac{5}{2} \le x \le p,\]</div><div class="list-item-paragraph">where&nbsp;\(p\) is a real constant.</div><ol class="i-list"><li><div class="list-item-paragraph">Find the largest exact value of \(p\) such that&nbsp;\(\mathrm{h}^{-1}\) exists.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">With the value of \(p\) obtained in part <strong>(b)(i)</strong>, find the solution of \(\mathrm{h}\mathrm{h}^{-1}(x) = \mathrm{h}^{-1}\mathrm{h}(x)\), leaving your answer to 3 decimal places.<span class="inline-marks">[2]</span></div></li></ol></li></ol>

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by

\[\mathrm{f} : x \mapsto \frac{\sqrt{2}(x-2)}{x^2+x-2}, \quad x \in \mathbb{R}, x > 1,\]

\[\mathrm{g} : x \mapsto \mathrm{e}^{-x}-1, \quad x \in \mathbb{R}.\]

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

\[\mathrm{h} : x \mapsto \sqrt{3}(\sin x + \cos x), \quad x \in \mathbb{R}, -\frac{5}{2} \le x \le p,\]

Functions

International Baccalaureate Higher Level Mathematics

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