2025 NYJC P1 Q9

The function \(\mathrm{f}\) is defined by

\[\mathrm{f}(x) = \frac{x + 1}{2x - 3} \text{ for}\hspace{0.5em} x \in \mathbb{R}, x \neq k.\]

1. State the value of \(k\) and explain why this value has to be excluded from the domain of \(\mathrm{f}\).[2]
2. Determine, with a reason, if \(\mathrm{f}^2\) exists.[2]
3. Find \(\mathrm{f}^{-1}(x)\).[2]
4. Hence find \(\mathrm{h}(x)\) for which \(\mathrm{fh}(x) = \frac{3x - 7}{2x - 1}, x \neq \frac{1}{2}\).[3]