(a) To find the inverse, start with \(y = (x + a)^2 - a\).

Rearrange to solve for \(x\):

\(y + a = (x + a)^2\)

\(x + a = \pm \sqrt{y + a}\)

Since \(x \leq -a\), choose the negative root:

\(x = -\sqrt{y + a} - a\)

Thus, \(f^{-1}(x) = -\sqrt{x + a} - a\).

(b)(i) The domain of \(f^{-1}\) is the range of \(f\), which is \(x > -a\).

(b)(ii) The range of \(f^{-1}\) is the domain of \(f\), which is \(y \leq -a\).