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Let's solve the problem step by step. We are asked to find the inverse function \( f^{-1}(x) \) for the given function \( f(x) = \sqrt[3]{x} + 4 \).

### Steps to Find the Inverse Function:

1. **Express the function**: Start with the equation \( y = \sqrt[3]{x} + 4 \).

2. **Swap \( x \) and \( y \)**: To find the inverse, we switch \( x \) and \( y \):
   \[
   x = \sqrt[3]{y} + 4
   \]

3. **Solve for \( y \)**: 
   - First, subtract 4 from both sides to isolate the cube root:
     \[
     x - 4 = \sqrt[3]{y}
     \]
   - Then, cube both sides to get rid of the cube root:
     \[
     (x - 4)^3 = y
     \]
   - So, the inverse function is:
     \[
     f^{-1}(x) = (x - 4)^3
     \]

### Answer:
The correct inverse function is \( f^{-1}(x) = (x - 4)^3 \).

Would you like more details on this or have any further questions?

Here are five related questions to consider:
1. How would you find the inverse of a quadratic function?
2. What is the importance of finding the domain and range of inverse functions?
3. Can a function always have an inverse? Why or why not?
4. What is the graphical relationship between a function and its inverse?
5. How do you verify that two functions are inverses of each other?

**Tip:** When finding the inverse of a function, remember that you essentially reverse the operations performed on the variable.

For the function f(x) = ³√x + 4, find f⁻¹(x).

Learn how to find the inverse function of f(x) = ³√x + 4 using step-by-step algebraic methods. This problem is ideal for students in Grades 9-11 covering key concepts in inverse functions and algebra.

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