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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 $ ?

With your answers can you write also some references?
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 $ ?

Thanks for answers.

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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 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
up vote 0 down vote accepted

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.

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I inclined to agree with Gauss too :P – Pierfrancesco PierQR Aiello Oct 7 '12 at 6:10

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