To find the smallest possible value of \(k\), we need the function \(h(x) = x^2 + 4x\) to be one-to-one, which occurs when it is either strictly increasing or decreasing. The vertex of the parabola \(x^2 + 4x\) is at \(x = -\frac{b}{2a} = -\frac{4}{2} = -2\). Thus, the smallest possible value of \(k\) is \(-2\) so that the function is increasing for \(x \geq -2\).

To find \(h^{-1}(x)\), start by setting \(y = x^2 + 4x\).

Complete the square: \(y = (x+2)^2 - 4\).

Rearrange to solve for \(x\):

\(y + 4 = (x+2)^2\)

\(\sqrt{y + 4} = x + 2\)

\(x = \sqrt{y + 4} - 2\)

Thus, the inverse function is \(h^{-1}(x) = \sqrt{x+4} - 2\).