(a(i))[1]
The one-one function $f$ is given by $f(x) = (x - 3)^2 - 1$ for $x < a$, where $a$ is a constant. State the largest possible value of $a$.
(a(ii))[3]
It is given that $a$ has this greatest possible value. State the range of $f$ and obtain an expression for $f^{-1}(x)$.
(b(i))[2]
The function $g$ is defined by $g(x) = (x - 3)^2$ for $x \geq 0$. Show that $g(g(2x))$ may be written in the form $(2x - 3)^4 + b(2x - 3)^2 + c$, where $b$ and $c$ are constants to be determined.
(b(ii))[4]
Hence expand $g(g(2x))$ fully, and simplify your answer.