2024 VJC Promo Q11

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

\(\mathrm{f} : x \mapsto \frac{5x + 13}{x^2 - 1} \quad\quad\quad \text{for}\hspace{0.5em} x \in \mathbb{R}, \; x \neq \pm 1,\)
\( \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \mathrm{g} : x \mapsto -\ln(x - 1) \quad \text{for}\hspace{0.5em} x \ge 2.\)

1. 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.[4]
2. Explain why \(\mathrm{g}\mathrm{f}\) does not exist.[1]
3. Show that \(\mathrm{g}\) has an inverse and find \(\mathrm{g}^{-1}(x)\).[4]
4. Find the range of \(\mathrm{f}\mathrm{g}^{-1}\).[2]