<p>The function \(\mathrm{f}\) is defined by \({f} : x \to 2x - \frac{1}{2x}, \, 0 &lt; x &lt; 2.\) It is given that \(\mathrm{f}^{-1}\) exists.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Define \(\mathrm{f}^{-1}\) in a similar form.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Sketch the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\) on the same diagram.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The region \(R\) is bounded by the curve \(y = \mathrm{f}^{-1}(x)\), \(y = x\) and the \(y\)-axis. Find the exact area of \(R\).</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Another function \(\mathrm{g}\) is defined by \({g} : x \to \begin{cases} \frac{3}{2} + \frac{3}{3x - 11} & \text{for } x \le 3, \\ \left| x - \frac{1}{3}x^2 \right| & \text{for } 3 &lt; x \le 5. \end{cases}\)<br> Show that the composite function \(\mathrm{gf}\) exists and find the range of \(\mathrm{gf}\).<span class="inline-marks">[2]</span></div></li></ol>

The function \(\mathrm{f}\) is defined by \({f} : x \to 2x - \frac{1}{2x}, \, 0 < x < 2.\) It is given that \(\mathrm{f}^{-1}\) exists.

Define \(\mathrm{f}^{-1}\) in a similar form.

Sketch the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\) on the same diagram.

The region \(R\) is bounded by the curve \(y = \mathrm{f}^{-1}(x)\), \(y = x\) and the \(y\)-axis. Find the exact area of \(R\).

Another function \(\mathrm{g}\) is defined by \\[\mathrm{g} : x \to \begin{cases} \frac{3}{2} + \frac{3}{3x - 11} & \text{for } x \le 3, \\ \left| x - \frac{1}{3}x^2 \right| & \text{for } 3 < x \le 5. \end{cases}\\] Show that the composite function \(\mathrm{gf}\) exists and find the range of \(\mathrm{gf}\).

Another function \(\mathrm{g}\) is defined by \({g} : x \to \begin{cases} \frac{3}{2} + \frac{3}{3x - 11} & \text{for } x \le 3, \\ \left| x - \frac{1}{3}x^2 \right| & \text{for } 3 < x \le 5. \end{cases}\)
 Show that the composite function \(\mathrm{gf}\) exists and find the range of \(\mathrm{gf}\).

2025 ASRJC P1 Q10

2025 ASRJC P1 Q10