<p>The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto \sqrt{5 + 2x}\), for&nbsp;\(x \in \mathbb{R}\), \(x &gt; -\frac{5}{2}\).</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of \(a\) such that&nbsp;\(\mathrm{f}^2(a) = 5\).<span class="inline-marks">[2]</span></div></li></ol>
<p>Another function \(\mathrm{g}\) is defined by \(\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|\), for&nbsp;\(x \in \mathbb{R}\), \(x \le k\).</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">State the largest value of \(k\) for&nbsp;\(\mathrm{g}^{-1}\) to exist.<span class="inline-marks">[1]</span></div></li></ol>
<p>It is now given that&nbsp;\(k = -1\).</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find \(\mathrm{g}^{-1}(x)\) and state its domain.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Explain why the composite function \(\mathrm{fg}\) exists. State the exact range of \(\mathrm{fg}\).<span class="inline-marks">[2]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by \(\mathrm{f} : x \mapsto \sqrt{5 + 2x}\), for \(x \in \mathbb{R}\), \(x > -\frac{5}{2}\).

Find the value of \(a\) such that \(\mathrm{f}^2(a) = 5\).

Another function \(\mathrm{g}\) is defined by \(\mathrm{g} : x \mapsto \left|\frac{x}{1 - 2x}\right|\), for \(x \in \mathbb{R}\), \(x \le k\).

State the largest value of \(k\) for \(\mathrm{g}^{-1}\) to exist.

Find \(\mathrm{g}^{-1}(x)\) and state its domain.

Explain why the composite function \(\mathrm{fg}\) exists. State the exact range of \(\mathrm{fg}\).

Functions

International Baccalaureate Higher Level Mathematics

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