To find the inverse of the function $f(x) = x^3 + 3$ algebraically, follow these steps:

1. Start with the function: $f(x) = x^3 + 3$ We aim to find $f^{-1}(x)$, the inverse function.

2. To do this, first replace $f(x)$ with $y$: $y = x^3 + 3$

3. Swap $x$ and $y$ to reflect that we are now solving for the inverse: $x = y^3 + 3$

4. Solve for $y$: $x - 3 = y^3$

   Now, take the cube root of both sides: $y = \sqrt[3]{x - 3}$

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

So, the inverse function of $f(x) = x^3 + 3$ is: $f^{-1}(x) = \sqrt[3]{x - 3}$

Let me know if you would like further explanation or have any other questions!

Related questions:

1. What is the inverse function of $f(x) = x^2 + 5$, and how does the process differ for non-cubic functions?
2. How do you verify if two functions are inverses of each other?
3. Can all functions have inverse functions? Why or why not?
4. How do transformations (e.g., shifts, stretches) affect the inverse of a function?
5. What are the general conditions for a function to be invertible?

Tip:

When finding the inverse of a function, always ensure the original function is one-to-one (bijective), as this guarantees the existence of an inverse function.