# Conventions for notation of function exponentation.

I read a previous question here but it seems incomplete for me (missing references).

Given a generic function, $f$ :
1. is true that $f^2$ means $f^2(x) = (f \circ f)(x) = f(f(x))$ ?
2. or is true that $f^2$ means $f^2(x) = (f(x))^2$ ?

Anyway if (2) holds, then is $(f \circ f)(x)$ the only way to write $f(f(x))$ ?
Instead, if (1) holds, then why do some books write $\ln^2(x) = (\ln(x))^2$ or the trigonometric identity $\sin^2(x)+\cos^2(x) = 1$ ?

-
It is often used in both ways (just not in the same context). You'll simply have to determine which is intended from the situation. – Cameron Buie Oct 6 '12 at 20:23
I'ld also suggest you to read the answer given here math.stackexchange.com/questions/30317/arcsin-written-as-sin-1x as it is quite well written – TheJoker Oct 6 '12 at 20:26
If you want an opinion on that, here is mine: Use $f^n(x) := (f(x))^n$ only if $f$ is a standard function. It is very common for trigonometric and hyperbolical functions, but that's about it. – filmor Oct 6 '12 at 20:28
thanks for answers. Filmor brought a good argument (even if i'm a Gauss fan here xD ) and now i'm reading the link about arcsin. – Pierfrancesco PierQR Aiello Oct 7 '12 at 6:32

I've seen both. I seem to recall having read somewhere (Therefore it's true! Right?) that Gauss objected to writing $\sin^2 x$ for $(\sin x)^2$ on the grounds that $\sin^2x$ ought to mean $\sin\sin x$. I'm inclined to agree with Gauss, but then there's King Canute and all that.
Certainly using $f^2$ to mean $f\circ f$ is consistent with the usual notation by which $f^{-1}$ means the inverse function.