Problem

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows, where $a$ and $b$ are constants.

$\mathrm{f}(x)=1+\frac{2a}{x-a}\quad\text{for }x>a$

$\mathrm{g}(x)=bx-2\quad\text{for }x\in\mathbb{R}$

(a) Given that $\mathrm{f}(7)=\frac{5}{2}$ and $\mathrm{gf}(5)=4$, find the values of $a$ and $b$.

For the rest of this question, you should use the value of $a$ which you found in (a).

(b) Find the domain of $\mathrm{f}^{-1}$.

(c) Find an expression for $\mathrm{f}^{-1}(x)$.