Problem

The function $f$ is defined by

$$f(x)=3+\frac{7}{x-2}$$

for $x>2$.

(a) It is given that $f(a)=4$.

Find the value of $a$.

(b) Find an expression for $f^{-1}(x)$ and state the domain of $f^{-1}$.

(c) The function $g$ is defined by

$$g(x)=\frac{1+4x}{2x-3}$$

for $x>\frac{3}{2}$.

Show that $fg(x)=kx$, where $k$ is a constant to be determined.