Problem

The function $f$ is defined by $f(x)=\frac{4}{(3x-6)^2}+\frac{1}{(3x-6)^3}$ for $x>2$.

(a) Find an expression for $f'(x)$ and hence determine whether $f$ is an increasing function, a decreasing function or neither.

(b) State whether $f^{-1}$ exists. Give a reason for your answer.

The function $g$ is defined by $g(x)=4x-3$ for $x>a$.

(c) Find the range of $g$ in terms of the constant $a$.

(d) Find the set of values of $a$ for which the composite function $fg$ exists.