2024 NYJC Promo Q7

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

\[\begin{aligned} \mathrm{f} & : x \mapsto |x + 20|, \quad x \in \mathbb{R}, \\ \mathrm{g} & : x \mapsto x + \frac{\alpha}{x}, \quad x \in \mathbb{R}, \ x > 0. \end{aligned}\]

1. Explain why the composite function \(\mathrm{gf}\) does not exist.[2]
2. Determine the range of values of \(\alpha\) such that \(\mathrm{g}^{-1}\) exists.[2]

For the rest of the question, it is given that \(\alpha = 4\) and the domain of \(\mathrm{g}\) is further restricted to \(x \ge \beta\).

3. Determine the least value of \(\beta\) for which \(\mathrm{g}^{-1}\) exists. Using this value of \(\beta\), hence find \(\mathrm{g}^{-1}(x)\) and state its domain.[4]