<p>The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by</p>
<p style="text-align: center">\(\mathrm{f} : x \mapsto \frac{5x + 13}{x^2 - 1} \quad\quad\quad \text{for}\hspace{0.5em} x \in \mathbb{R}, \; x \neq \pm 1,\)<br>\( \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mathrm{g} : x \mapsto -\ln(x - 1) \quad \text{for}\hspace{0.5em} x \ge 2.\)</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Sketch the graph of \(y = \mathrm{f}(x)\), stating the equations of any asymptotes, the coordinates of any turning points and points of intersections with the axes.<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Explain why \(\mathrm{g}\mathrm{f}\) does not exist.<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">Show that&nbsp;\(\mathrm{g}\) has an inverse and find \(\mathrm{g}^{-1}(x)\).<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Find the range of \(\mathrm{f}\mathrm{g}^{-1}\).<span class="inline-marks">[2]</span></div></li></ol>

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

\(\mathrm{f} : x \mapsto \frac{5x + 13}{x^2 - 1} \quad\quad\quad \text{for } x \in \mathbb{R}, \; x \neq \pm 1,\)
\( \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mathrm{g} : x \mapsto -\ln(x - 1) \quad \text{for } x \ge 2.\)

Sketch the graph of \(y = \mathrm{f}(x)\), stating the equations of any asymptotes, the coordinates of any turning points and points of intersections with the axes.

Explain why \(\mathrm{g}\mathrm{f}\) does not exist.

Show that \(\mathrm{g}\) has an inverse and find \(\mathrm{g}^{-1}(x)\).

Find the range of \(\mathrm{f}\mathrm{g}^{-1}\).

2024 VJC Promo Q11

2024 VJC Promo Q11