1. **State the problem:** Find the inverse function of $f(x) = x^3 - 12$. 2. **Recall the definition of inverse function:** The inverse function $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$. 3. **Set $y = f(x)$:** $$y = x^3 - 12$$ 4. **Swap $x$ and $y$ to find the inverse:** $$x = y^3 - 12$$ 5. **Solve for $y$:** $$x + 12 = y^3$$ 6. **Take the cube root of both sides:** $$y = \sqrt[3]{x + 12}$$ 7. **Therefore, the inverse function is:** $$f^{-1}(x) = \sqrt[3]{x + 12}$$ 8. **Note:** The original function is a cubic, so its inverse is a cube root function, not a square root function. The expression $\sqrt{x + [\ ]}$ with a plus sign inside the root does not represent the inverse of this cubic function. **Final answer:** $$f^{-1}(x) = \sqrt[3]{x + 12}$$