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To find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \sqrt[3]{\frac{x - 5}{9}} \), we will follow these steps:

1. **Rewrite the function as an equation:**
   \[
   y = \sqrt[3]{\frac{x - 5}{9}}
   \]

2. **Swap \( x \) and \( y \) to find the inverse:**
   \[
   x = \sqrt[3]{\frac{y - 5}{9}}
   \]

3. **Cube both sides to eliminate the cube root:**
   \[
   x^3 = \frac{y - 5}{9}
   \]

4. **Multiply both sides by 9:**
   \[
   9x^3 = y - 5
   \]

5. **Add 5 to both sides to solve for \( y \):**
   \[
   y = 9x^3 + 5
   \]

So, the inverse function is:
\[
f^{-1}(x) = 9x^3 + 5
\]

Do you want details or have any questions? Here are five more questions you can ask:

1. How do you find the domain and range of an inverse function?
2. Can you show an example of verifying an inverse function?
3. What are some real-life applications of inverse functions?
4. How do you differentiate inverse trigonometric functions?
5. Can you explain the horizontal line test for determining if a function has an inverse?

**Tip:** When dealing with roots and exponents in functions, carefully apply the necessary algebraic operations to isolate the variable when finding the inverse.

Discover how to find the inverse function f^{-1}(x) for f(x) = cube root of ((x - 5) / 9) step-by-step. This problem involves functions, inverse functions, and cube root functions, suitable for advanced high school students (Grades 11-12).

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