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To find the inverse of the function \( f(x) = \sqrt[3]{x + 5} + 1 \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \):**
   \[
   y = \sqrt[3]{x + 5} + 1
   \]

2. **Solve for \( x \) in terms of \( y \):**
   \[
   y - 1 = \sqrt[3]{x + 5}
   \]
   \[
   (y - 1)^3 = x + 5
   \]
   \[
   x = (y - 1)^3 - 5
   \]

3. **Replace \( y \) with \( x \) to express the inverse function:**
   \[
   f^{-1}(x) = (x - 1)^3 - 5
   \]

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

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

Here are 5 questions you might want to ask next:
1. How do you verify that two functions are inverses of each other?
2. Can you explain the steps to find the derivative of \( f(x) \)?
3. What is the domain and range of \( f(x) \) and its inverse?
4. How do you find the composition of functions?
5. What is the importance of inverse functions in calculus?

**Tip:** Always check if the function is one-to-one before finding its inverse to ensure the inverse function exists.

Learn how to find the inverse function of f(x) = cube root of (x + 5) + 1 with a step-by-step solution. This problem involves inverse functions, cube roots, and algebraic manipulation. Suitable for advanced high school students in Grades 11-12.

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