To find the inverse of the function $f(x) = x^2 + 4$, follow these steps:

1. Express $f(x)$ as $y$: $y = x^2 + 4$

2. Swap $x$ and $y$ to find the inverse: $x = y^2 + 4$

3. Solve for $y$: $x - 4 = y^2$ $y = \pm\sqrt{x - 4}$

Since the original function $f(x) = x^2 + 4$ is not one-to-one over the entire set of real numbers (it fails the horizontal line test), its inverse will not be a function unless we restrict the domain. Typically, we restrict the domain to $x \ge 0$ or $x \le 0$. Let's consider the restriction $x \ge 0$:

4. With the restriction $x \ge 0$, the inverse function is: $f^{-1}(x) = \sqrt{x - 4}$

For completeness, if the domain of $f(x)$ were restricted to $x \le 0$, the inverse function would be: $f^{-1}(x) = -\sqrt{x - 4}$

Thus, the inverse function of $f(x) = x^2 + 4$ is: $f^{-1}(x) = \sqrt{x - 4}$ if $x \ge 0$.

If you have any questions or need further clarification, let me know! Here are five more questions you might consider:

1. How do you determine if a function has an inverse?
2. What are the properties of inverse functions?
3. How do you find the inverse of a function involving radicals?
4. How do domain restrictions affect the inverse of a function?
5. What are some common mistakes to avoid when finding the inverse of a function?

Tip: Always check if the function is one-to-one (passes the horizontal line test) before finding its inverse. If not, consider restricting the domain.