To find the inverse of the function $f(x) = \sqrt[3]{x + 7} + 5$, follow these steps:

Step 1: Replace $f(x)$ with $y$

$y = \sqrt[3]{x + 7} + 5$

Step 2: Solve for $x$ in terms of $y$

1. Subtract 5 from both sides: $y - 5 = \sqrt[3]{x + 7}$

2. Cube both sides to eliminate the cube root: $(y - 5)^3 = x + 7$

3. Finally, subtract 7 from both sides: $x = (y - 5)^3 - 7$

Step 3: Replace $x$ with $f^{-1}(x)$ and $y$ with $x$

Thus, the inverse function is: $f^{-1}(x) = (x - 5)^3 - 7$

Final Answer:

$f^{-1}(x) = (x - 5)^3 - 7$

Would you like further clarification on any step, or have any additional questions?

Here are some related questions:

1. How do we graph the original function and its inverse?
2. What is the domain and range of the function and its inverse?
3. How do we verify that two functions are inverses of each other?
4. How do cube roots affect the shape of a graph compared to square roots?
5. What happens to the inverse of a function when there are transformations such as translations or scalings?

Tip: When finding the inverse of a function, always swap the roles of $x$ and $y$ and solve for the new $y$.