Question: User: f(x) = ³√(x + 10). Find the inverse of f(x).
A. f^-1(x) = x^3 - 10
B. f^-1(x) = (x - 10)^3
C. f^-1(x) = x^3 + 10
D. f^-1(x) = (x + 10)^3
1. **State the problem:** We are given the function $$f(x) = \sqrt[3]{x + 10}$$ and asked to find its inverse function $$f^{-1}(x)$$.
2. **Recall the definition of inverse function:** The inverse function $$f^{-1}(x)$$ satisfies $$f(f^{-1}(x)) = x$$ and $$f^{-1}(f(x)) = x$$.
3. **Set up the equation:** Let $$y = f(x) = \sqrt[3]{x + 10}$$.
4. **Express $$x$$ in terms of $$y$$:**
$$y = \sqrt[3]{x + 10}$$
Cube both sides to eliminate the cube root:
$$y^3 = x + 10$$
5. **Solve for $$x$$:**
$$x = y^3 - 10$$
6. **Interchange variables to write inverse function:**
Replace $$y$$ by $$x$$ to get the inverse function:
$$f^{-1}(x) = x^3 - 10$$
7. **Conclusion:** The inverse function is $$f^{-1}(x) = x^3 - 10$$, which corresponds to option A.
**Final answer:** $$\boxed{f^{-1}(x) = x^3 - 10}$$