To find the inverse of the function \( f(x) = \sqrt[3]{x + 12} \), we need to follow these steps:
1. Rewrite the function as an equation with \( y \) and \( x \):
Let \( y = f(x) \). So, we have:
[tex]\[
y = \sqrt[3]{x + 12}
\][/tex]
2. Switch the variables \( x \) and \( y \), reflecting the fact that we want the input and output exchanged:
[tex]\[
x = \sqrt[3]{y + 12}
\][/tex]
3. Eliminate the cube root by cubing both sides of the equation:
[tex]\[
x^3 = y + 12
\][/tex]
4. Solve for \( y \) to isolate it:
Subtract 12 from both sides:
[tex]\[
y = x^3 - 12
\][/tex]
So, the inverse function is:
[tex]\[
f^{-1}(x) = x^3 - 12
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
