[Math] How to describe the growth of something that scales by a factorial

Question

I just wrote a blog post and wasn't sure how to word a particular sentence. Say I have the following function:

\begin{equation}
f(x) = x^2
\end{equation}

Then I can say that the value of f(x) grows quadratically with x*. Similarly with this function:

\begin{equation}
f(x) = e^x
\end{equation}

…I could say that f(x) grows exponentially with x. But what about this?

\begin{equation}
f(x) = x!
\end{equation}

Do I say that f(x) grows "factorially"? What's the proper term?

---

_{*I worded this wrong at first. It should be right now. Wait, is that even right? Or would "exponentially" imply f(x) = k^x? Should the first term be "quadratically" instead?}

Answer 1

Using Stirling's formula,

$$n! \times e^n \approx Cn^{n + 1/2}$$

I am not sure if there is a name for that kind of growth. It is super-exponential and might be enough to get the point across, I suppose.

btw, $f(x) = x^2$ is said to grow quadratically, not exponentially.