(i) (a) For \(k = 2\), \(f(x) = 3 - 4 \cos^2 x\). The maximum value of \(\cos^2 x\) is 1, and the minimum is 0. Thus, the range of \(f(x)\) is from \(3 - 4(1) = -1\) to \(3 - 4(0) = 3\). Therefore, the range is \([-1, 3]\).

(b) To solve \(f(x) = 1\), set \(3 - 4 \cos^2 x = 1\). This simplifies to \(\cos^2 x = \frac{1}{2}\), giving \(\cos x = \pm \frac{1}{\sqrt{2}}\). The solutions in the interval \(0 \leq x \leq \pi\) are \(x = \frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\).

(ii) (a) For \(k = 1\), \(f(x) = 3 - 4 \cos x\). The function is increasing from \((0, -1)\) to \((\pi, 7)\). It is not a straight line and flattens at the extremities, indicating an inflection.

(b) The function \(f\) has an inverse because it is one-to-one (1:1) or strictly increasing with no turning points.