<p>The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by</p>
<p>\[\begin{aligned} &\mathrm{f} : x \mapsto 5 - (x - 2)^2, \quad \text{for}\hspace{0.5em} x \in \mathbb{R}, \ x \ge 0, \\ &\mathrm{g} : x \mapsto \mathrm{e}^x + 3, \qquad \text{for}\hspace{0.5em} x \in \mathbb{R}. \end{aligned}\]</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that the composite function \(\mathrm{fg}\) exists.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find an expression for&nbsp;\(\mathrm{fg}(x)\) and state its domain.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Find the range of \(\mathrm{fg}\).<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;\(\mathrm{f}^{-1}\) does not exist.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">If the domain of \(\mathrm{f}\) is restricted to \([k, \infty)\), find the smallest value of \(k\) such that&nbsp;\(\mathrm{f}^{-1}\) exists.<span class="inline-marks">[1]</span></div></li></ol>
<p>For the rest of the question, the domain of \(\mathrm{f}\) is now restricted to \([k, \infty)\), with the value of \(k\) found in part <strong>(v)</strong>.</p>
<ol class="i-list" start="6"><li><div class="list-item-paragraph">Find \(\mathrm{f}^{-1}(x)\) and state the domain of \(\mathrm{f}^{-1}\).<span class="inline-marks">[3]</span></div></li></ol>

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

\[\begin{aligned} &\mathrm{f} : x \mapsto 5 - (x - 2)^2, \quad \text{for } x \in \mathbb{R}, \ x \ge 0, \\ &\mathrm{g} : x \mapsto \mathrm{e}^x + 3, \qquad \text{for } x \in \mathbb{R}. \end{aligned}\]

Show that the composite function \(\mathrm{fg}\) exists.

Find an expression for \(\mathrm{fg}(x)\) and state its domain.

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

If the domain of \(\mathrm{f}\) is restricted to \([k, \infty)\), find the smallest value of \(k\) such that \(\mathrm{f}^{-1}\) exists.

For the rest of the question, the domain of \(\mathrm{f}\) is now restricted to \([k, \infty)\), with the value of \(k\) found in part **(v)**.

Find \(\mathrm{f}^{-1}(x)\) and state the domain of \(\mathrm{f}^{-1}\).

2024 MI PRU1 Q7

2024 MI PRU1 Q7