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We are given the function \( f(x) = 4 + \sqrt[3]{x} \), and we need to find its inverse, \( f^{-1}(x) \).

### Steps to find the inverse function:

1. Start by replacing \( f(x) \) with \( y \) for simplicity:
   \[
   y = 4 + \sqrt[3]{x}
   \]
   
2. Switch \( x \) and \( y \) to begin solving for \( y \) (this is the inverse step):
   \[
   x = 4 + \sqrt[3]{y}
   \]
   
3. Isolate the cube root term by subtracting 4 from both sides:
   \[
   x - 4 = \sqrt[3]{y}
   \]
   
4. Cube both sides to eliminate the cube root:
   \[
   (x - 4)^3 = y
   \]
   
5. Thus, the inverse function is:
   \[
   f^{-1}(x) = (x - 4)^3
   \]

The inverse of \( f(x) = 4 + \sqrt[3]{x} \) is \( f^{-1}(x) = (x - 4)^3 \).

Would you like further details or have any questions about this solution?

Here are 5 related questions:
1. How do you verify if two functions are inverses of each other?
2. Can every function have an inverse?
3. What is the domain and range of the inverse function \( f^{-1}(x) \)?
4. How does the transformation of shifting and cubing affect the graph of a function?
5. What role does the cube root play in finding the inverse?

**Tip:** When finding an inverse function, always switch \( x \) and \( y \), then solve for \( y \), which gives the inverse.

Find the inverse function of f(x) = 4 + ³√x.

Learn how to find the inverse function of f(x) = 4 + ³√x in this detailed algebra solution. Follow step-by-step instructions to derive the inverse function f⁻¹(x) = (x - 4)³, covering key concepts in algebra such as inverse functions and cube roots. Suitable for students in Grades 9-12.

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