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.