# How do I describe the growth of something that scales by a factorial?

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

$$f(x) = x^2$$

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

$$f(x) = e^x$$

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

$$f(x) = x!$$

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) = kx? Should the first term be "quadratically" instead?

• $f(x) = x^2$ grows quadratically with $x$. $f(x) = e^x$ grows exponentially. Jan 23, 2011 at 22:38
• @Moron: Thanks, you pointed that out just as I started to realize my mistake (see my edits—I've now corrected that part). Jan 23, 2011 at 22:40
• I think you mean $e^x$ where you wrote $e^2$.
– user856
Jan 23, 2011 at 23:03
• @Rahul: Yes I did, thanks for pointing that out! (Man, clearly it pays to proofread your math.) Jan 23, 2011 at 23:04

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.

• @Moron: You forgot about the $e^{-n}$-term in Stirling's formula. Jan 23, 2011 at 22:45
• @Rasmus: Yes, was editing :-)Thanks for pointing that out. Jan 23, 2011 at 22:46
• Well, $n^n$ can be described as tetration, in which case it could be denoted ${}^2n$. Not sure how that word could be turned into an adjective though, and it's not exactly widely used I guess :) Jan 23, 2011 at 23:13
• @Will: Yes, but $n!$ is $n^{n+1/2}$ divided by $e^n$, so it is not exactly tetration... There is also that $\sqrt{n}$ factor. Jan 24, 2011 at 0:38
• I'd say this, combined with Will's comment, answers my question. Thanks! Jan 24, 2011 at 2:56