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We are given the one-to-one function \( f(x) = \sqrt[3]{1 - x} \), and we need to find its inverse, \( f^{-1}(x) \).

To find the inverse of a function, we follow these steps:

### Steps to find \( f^{-1}(x) \):

1. **Rewrite the function with \( y = f(x) \):**

   \[
   y = \sqrt[3]{1 - x}
   \]

2. **Swap \( x \) and \( y \):**

   \[
   x = \sqrt[3]{1 - y}
   \]

3. **Solve for \( y \):**
   - Cube both sides to eliminate the cube root:

   \[
   x^3 = 1 - y
   \]

   - Solve for \( y \):

   \[
   y = 1 - x^3
   \]

Thus, the inverse function is:

\[
f^{-1}(x) = 1 - x^3
\]

### Final Answer:
\[
f^{-1}(x) = 1 - x^3
\]

Would you like further clarification or additional details on any of the steps?

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Here are some related questions to explore further:
1. What is the domain and range of \( f(x) \) and its inverse?
2. How would you verify that \( f(x) \) and \( f^{-1}(x) \) are indeed inverses?
3. How does the graph of \( f(x) \) relate to the graph of \( f^{-1}(x) \)?
4. What are the general conditions for a function to have an inverse?
5. How does the cubic root function affect the shape of the graph?

**Tip:** When finding the inverse of a function, always make sure that the original function is one-to-one (injective), which guarantees that its inverse exists.

Given the one-to-one function f(x) = ∛(1 - x), find f^{-1}(x).

Learn how to find the inverse function f^{-1}(x) for the given one-to-one function f(x) = ∛(1 - x). This problem covers important algebraic concepts such as inverse functions, cube roots, and step-by-step algebraic manipulations. Suitable for high school students in Grades 10-12.

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