9758/2022/P1/Q06

The function $\mathrm{f}$ is defined by
$$f: x \rightarrow \frac{a x+k}{x-a}, \quad x \in \mathbb{R}, \quad x \neq a$$
where $a$ and $k$ are constants.
(a) Describe fully a sequence of transformations which transforms the curve $y=\frac{1}{x}$ onto the curve $y=\mathrm{f}(x)$.

[4]

(b) Find $\mathrm{f}^{-1}(x)$.

[2]

(c) Hence, or otherwise, find $\mathrm{f}^2(x)$.

[1]

(d) Find $\mathrm{f}^{2023}(1)$ in terms of $a$ and $k$.

[2]