2025 TMJC P1 Q7

The function \(\mathrm{f}\) is such that\[\mathrm{f} : x \mapsto \frac{1}{2} \sqrt{36 - (x - 3)^2} \text{ for}\hspace{0.5em} x \in \mathbb{R}, \, k \leq x < 9,\]where \(k\) is a real constant.

1. State the least value of \(k\) for which the function \(\mathrm{f}^{-1}\) exists.[1]

For the rest of the question, take the value of \(k\) as the value found in part (a).

2. Find \(\mathrm{f}^{-1}(x)\).[3]
3. Sketch, on the same diagram, the graphs of \(y = \mathrm{f}(x)\), \(y = \mathrm{f}^{-1}(x)\) and \(y = \mathrm{f}^{-1}\mathrm{f}(x)\), showing clearly the relationship between the three graphs.[3]

The function \(\mathrm{g}\) is such that\[\mathrm{g} : x \mapsto |4x - 1|(2x - 5) \text{ for}\hspace{0.5em} x \in \mathbb{R}, \, x > 0.\]

4. Show that the composite function \(\mathrm{gf}\) exists.[2]
5. Find the range of \(\mathrm{gf}\).[2]