1. **State the problem:** We are given the function $$f(x) = (x+1)^4 (x-2)^5$$ and asked to find its inverse function $$f^{-1}(x)$$. 2. **Understand the problem:** Finding the inverse function means solving for $$x$$ in terms of $$y$$ where $$y = f(x)$$. 3. **Write the equation:** $$y = (x+1)^4 (x-2)^5$$. 4. **Analyze the function:** This is a product of two polynomial terms raised to powers 4 and 5. The function is not one-to-one over all real numbers because it is a polynomial of degree 9, which is generally not invertible by elementary algebraic methods. 5. **Conclusion:** There is no simple closed-form expression for $$f^{-1}(x)$$ using elementary functions because solving $$y = (x+1)^4 (x-2)^5$$ for $$x$$ involves solving a ninth-degree polynomial equation. 6. **Alternative:** The inverse function can be expressed implicitly or approximated numerically but cannot be written explicitly in terms of elementary functions. **Final answer:** The inverse function $$f^{-1}(x)$$ cannot be expressed in a closed form using elementary functions.