To find the inverse function $f^{-1}(x)$ for the given function $f(x) = 10 + \sqrt[3]{x}$, follow these steps:

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

Let: $y = 10 + \sqrt[3]{x}$

Step 2: Swap $x$ and $y$

For the inverse function, switch the roles of $x$ and $y$: $x = 10 + \sqrt[3]{y}$

Step 3: Solve for $y$

Isolate $y$: $x - 10 = \sqrt[3]{y}$ Now, cube both sides to eliminate the cube root: $(x - 10)^3 = y$

Thus, the inverse function is: $f^{-1}(x) = (x - 10)^3$

Would you like further clarification or details?

Here are some related questions to practice:

1. What is the domain and range of the inverse function $f^{-1}(x)$?
2. How would you verify if the inverse function is correct?
3. What is the inverse of $f(x) = 10 + \sqrt[4]{x}$?
4. Can a function always have an inverse? Why or why not?
5. How would you graph the function $f(x)$ and its inverse?

Tip: To find the inverse of a function, always swap $x$ and $y$, then solve for $y$.