<p>Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by</p>
<p style="text-align: center">\[\begin{aligned} \mathrm{f} & : x \mapsto |x + 20|, \quad x \in \mathbb{R}, \\ \mathrm{g} & : x \mapsto x + \frac{\alpha}{x}, \quad x \in \mathbb{R}, \ x &gt; 0. \end{aligned}\]</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Explain why the composite function \(\mathrm{gf}\) does not exist.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Determine the range of values of \(\alpha\) such that&nbsp;\(\mathrm{g}^{-1}\) exists.<span class="inline-marks">[2]</span></div></li></ol>
<p>For the rest of the question, it is given that&nbsp;\(\alpha = 4\) and the domain of \(\mathrm{g}\) is further restricted to \(x \ge \beta\).</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Determine the least value of \(\beta\) for which \(\mathrm{g}^{-1}\) exists. Using this value of \(\beta\), hence find \(\mathrm{g}^{-1}(x)\) and state its domain.<span class="inline-marks">[4]</span></div></li></ol>

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

\[\begin{aligned} \mathrm{f} & : x \mapsto |x + 20|, \quad x \in \mathbb{R}, \\ \mathrm{g} & : x \mapsto x + \frac{\alpha}{x}, \quad x \in \mathbb{R}, \ x > 0. \end{aligned}\]

Explain why the composite function \(\mathrm{gf}\) does not exist.

Determine the range of values of \(\alpha\) such that \(\mathrm{g}^{-1}\) exists.

For the rest of the question, it is given that \(\alpha = 4\) and the domain of \(\mathrm{g}\) is further restricted to \(x \ge \beta\).

Determine the least value of \(\beta\) for which \(\mathrm{g}^{-1}\) exists. Using this value of \(\beta\), hence find \(\mathrm{g}^{-1}(x)\) and state its domain.

2024 NYJC Promo Q7

2024 NYJC Promo Q7