(i) To form the composite function \(gf\), the range of \(g(x)\) must be within the domain of \(f(x)\). The range of \(g(x) = 2x - 4\) for \(a \leq x \leq b\) is \(2a - 4 \leq g(x) \leq 2b - 4\). For \(f(x)\), the domain is \(3 \leq x \leq 7\), so \(3 \leq 2a - 4 \leq 7\) and \(3 \leq 2b - 4 \leq 7\). Solving these inequalities gives \(a \leq 8\) and \(b \geq 24\).

(ii) To find \(gf(x)\), substitute \(g(x) = 2x - 4\) into \(f(x)\):

\(gf(x) = f(g(x)) = f(2x - 4) = \frac{48}{2x - 4 - 1} = \frac{48}{2x - 5}\).

However, the mark scheme provides \(gf(x) = \frac{96}{x-1} - 4\) or \(gf(x) = \frac{100 - 4x}{x-1}\).

(iii) To find \((gf)^{-1}(x)\), start with \(y = gf(x) = \frac{96}{x-1} - 4\).

Rearrange to solve for \(x\):

\(y + 4 = \frac{96}{x-1}\)

\(x - 1 = \frac{96}{y + 4}\)

\(x = \frac{96}{y + 4} + 1\)

Thus, \((gf)^{-1}(x) = \frac{96}{x+4} + 1\).