<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-paragraph">The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto x - 2 \mathrm{tan}^{-1}(2x)\), \(-\frac{\sqrt{3}}{2} &lt; x &lt; \frac{\sqrt{3}}{2}\).</div><ol class="i-list"><li data-list-item-id="item-a-i"><div class="list-item-paragraph">Show that&nbsp;\(\mathrm{f}(x)\) decreases as&nbsp;\(x\) increases.<span class="inline-marks">[3]</span></div></li><li data-list-item-id="item-a-ii"><div class="list-item-paragraph">Find the range of \(\mathrm{f}\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-a-iii"><div class="list-item-paragraph">Explain whether the composite function \(\mathrm{ff}\) exists.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-a-iv"><div class="list-item-paragraph">The curve \(C\) is obtained by reflecting the graph of \(\mathrm{f}\) in the line \(y = x\). Write down the equation of \(C\).<span class="inline-marks">[1]</span></div></li></ol></li><li data-list-item-id="item-b"><div class="list-item-paragraph">The function \(\mathrm{h}\) is defined by \(\mathrm{h} : x \mapsto x^2 + 2\), \(x \in \mathbb{R}\). The function \(\mathrm{g}\) is such that the composite function \(\mathrm{gh}\) exists and \[\mathrm{gh}(x) = x^4 - 2x^2 + 5, \quad x \in \mathbb{R}.\] Find an expression for&nbsp;\(\mathrm{g}(x)\).<span class="inline-marks">[2]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto x - 2 \mathrm{tan}^{-1}(2x)\), \(-\frac{\sqrt{3}}{2} < x < \frac{\sqrt{3}}{2}\).

The function \(\mathrm{h}\) is defined by \(\mathrm{h} : x \mapsto x^2 + 2\), \(x \in \mathbb{R}\). The function \(\mathrm{g}\) is such that the composite function \(\mathrm{gh}\) exists and \[\mathrm{gh}(x) = x^4 - 2x^2 + 5, \quad x \in \mathbb{R}.\] Find an expression for \(\mathrm{g}(x)\).

2009 IJC P1 Q10

2009 IJC P1 Q10