The function
\[
f(x)=-3 \sqrt{x+14}-12 \quad x \geq-14
\]
has an inverse $\mathrm{f}^{-1}(\mathrm{x})$ defined on the domain $\mathrm{x} \geq-12$. Find the inverse.

Step 1 ：Replace the function notation \(f(x)\) with \(y\), so we have: \(y = -3\sqrt{x+14} - 12\)

Step 2 ：Swap the roles of \(x\) and \(y\). This means we replace every \(x\) in the equation with \(y\) and every \(y\) with \(x\), giving us: \(x = -3\sqrt{y+14} - 12\)

Step 3 ：Add 12 to both sides of the equation: \(x + 12 = -3\sqrt{y+14}\)

Step 4 ：Divide both sides by -3: \(-\frac{x + 12}{3} = \sqrt{y+14}\)

Step 5 ：Square both sides to get rid of the square root: \(\left(-\frac{x + 12}{3}\right)^2 = y + 14\)

Step 6 ：Simplify the left side: \(\frac{x^2 + 24x + 144}{9} = y + 14\)

Step 7 ：Subtract 14 from both sides to solve for \(y\): \(y = \frac{x^2 + 24x + 144}{9} - 14\)

Step 8 ：\(\boxed{f^{-1}(x) = \frac{x^2 + 24x + 144}{9} - 14}\) is the inverse function, which is defined for \(x \geq -12\)