(a) The one-one function \(f\) is defined by \(f(x) = (x - 3)^2 - 1\) for \(x < a\), where \(a\) is a constant.

(i) State the greatest possible value of \(a\).

(ii) It is given that \(a\) takes this greatest possible value. State the range of \(f\) and find an expression for \(f^{-1}(x)\).

(b) The function \(g\) is defined by \(g(x) = (x - 3)^2\) for \(x \geq 0\).

(i) Show that \(gg(2x)\) can be expressed in the form \((2x - 3)^4 + b(2x - 3)^2 + c\), where \(b\) and \(c\) are constants to be found.

(ii) Hence expand \(gg(2x)\) completely, simplifying your answer.