<p>The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\rightarrow x\left|x-3\right|,\ x\in\mathbb{R},\ x&lt;m\).</p>
<ol class="i-list"><li data-list-item-id="item-i"><div class="list-item-paragraph">Show that&nbsp;\(\mathrm{f}^{-1}\) does not exist if&nbsp;\(m=4\).<span class="inline-marks">[1]</span></div></li><li data-list-item-id="item-ii"><div class="list-item-paragraph">State the largest possible value of \(m\) for which \(\mathrm{f}^{-1}\) exists.<span class="inline-marks">[1]</span></div></li></ol>
<p>Use the value of \(m\) obtained in part <strong>(ii)</strong> for the following questions.</p>
<ol class="i-list" start="3"><li data-list-item-id="item-iii"><div class="list-item-paragraph">Find \(\mathrm{f}^{-1}\), stating its domain.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-iv"><div class="list-item-paragraph">The composite function \(\mathrm{fg}\) is defined by \(\mathrm{fg}:x\rightarrow \frac{9}{4}-x^2,\ x\in\mathbb{R},\ x&lt;0\).</div><div class="list-item-paragraph">Find \(\mathrm{g}(x)\).<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-v"><div class="list-item-paragraph">The function \(\mathrm{h}\) is defined by \(\mathrm{h}:x\rightarrow \frac{3}{x-a},\ x\in\mathbb{R},\ x&lt;a\).</div><ol class="abc-list"><li data-list-item-id="item-v-a"><div class="list-item-paragraph">The graph of \(y=\mathrm{k}(x)\) is transformed by a stretch with scale factor 2 parallel to the \(x\)-axis, followed by a translation of 10 units in the negative \(y\)-direction. The equation of the resulting curve is \(y=\mathrm{h}(x)\).</div><div class="list-item-paragraph">Find \(\mathrm{k}(x)\).<span class="inline-marks">[2]</span></div></li></ol></li></ol>

The function \(\mathrm{f}\) is defined by \(\mathrm{f}:x\rightarrow x\left|x-3\right|,\ x\in\mathbb{R},\ x<m\).

Show that \(\mathrm{f}^{-1}\) does not exist if \(m=4\).

State the largest possible value of \(m\) for which \(\mathrm{f}^{-1}\) exists.

Use the value of \(m\) obtained in part **(ii)** for the following questions.

Find \(\mathrm{f}^{-1}\), stating its domain.

The composite function \(\mathrm{fg}\) is defined by \(\mathrm{fg}:x\rightarrow \frac{9}{4}-x^2,\ x\in\mathbb{R},\ x<0\).

The function \(\mathrm{h}\) is defined by \(\mathrm{h}:x\rightarrow \frac{3}{x-a},\ x\in\mathbb{R},\ x<a\).

2014 HCI P2 Q4 (part)

2014 HCI P2 Q4 (part)