How to denote "powers" of a function? I'm working with functions themselves, and I have learned that functional powers mean composition so:
$f^3 = f \circ f \circ f$
But I'm looking for something that means $fff$. So $(fff)(x) = (f(x))^3$
Is there a de facto standard notation I can use when mixing the two? Preferably without having to write out the $(x)$ part?
It seems we have a good deal ambiguity here. Right at the question title: The power of a function may mean two things. Sometimes even the circle is omitted to make it look like multiplication.
 A: The most common notation I've seen for $n$-fold composition is $$f(f(\ldots f(x)\ldots ))=f^{n}(x)$$
However this is generally always accompanied by a remark explaining that this is what the notation means. I would recommend you include such a remark.
I'm pretty sure there isn't a standard notation for raking a function to the power $n $, but again if you define some notation in the text then you're unlikely to be criticised.
A: FWIW, $f\,f\,f$ is also not quite unabiguous: it could be read as $f(f(f))$, or $f\:(f,f)$ – though both are really a bit strange and only make sense for polymorphic / higher-order functions.
Do use $f^n$ if you need this a lot, but also make some quick clarification about it. Or use something completely different – how about $\prod^n f$? That should be pretty unambiguous (at least unless you also have a symbol $\Pi$ around...).
A: What you seem to want is the composition of a function with cubing.
Although a little verbose, I think this is correct:
$$
(x \mapsto x^3) \circ f
$$
I don't think $^3 \circ f$ or $^3f$ is standard notation, but you may find it useful if you are repeating it a lot.
A: I've already seen the following, which I adopted: $$\begin{align} f^n(x) &= (f(x))^n \\ f^{(n)}(x) &= \frac{{\rm d}^nf}{{\rm d}x^n} \\ f^{\circ n}(x) &= (f \circ f \circ \cdots \circ f)(x)\end{align}$$
However $f^{\circ n}$ does not seem to be standard, so you should always warn the reader when using it.


The power of a function may mean two things. Sometimes even the circle is omitted to make it look like multiplication.

That's because the collection of all bijections from a set to itself form a group with composition of functions - and the operation in a group is usually seen as multiplication.
A: This is a negative consequence of the pigeon-hole principal. There are more mathematical concepts to express than there are (uncomplicated) symbols. There has to be some trade off between clarity and simplicity.
In 'Surely You're Joking Mr. Feynman', Richard Feynman talks about studying trigonometry.
"While I was doing all this trigonometry, I didn't like the symbols for sine, cosine, tangent, and so on. To me, "sin f" looked like s times i times n times f! So I invented another symbol, like a square root sign, that was a sigma with a long arm sticking out of it, and I put the f underneath. For the tangent it was a tau with the top of the tau extended, and for the cosine I made a kind of gamma, but it looked a little bit like the square root sign. Now the inverse sine was the same sigma, but left -to-right reflected so that it started with the horizontal line with the value underneath, and then the sigma. That was the inverse sine, NOT sink f--that was crazy! They had that in books! To me, sin_i meant i/sine, the reciprocal. So my symbols were better."
$\cdots$
I thought my symbols were just as good, if not better, than the regular symbols--it doesn't make any difference what symbols you use--but I
discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to
make these symbols, and he said, "What the hell are those?" I realized then that if I'm going to talk to anybody else, I'll have to use the standard
symbols, so I eventually gave up my own symbols.
A: As far as I am aware notation $f^n$ would only be taken to mean repeated (point-wise) multiplication of a function with itself without further mention if your setting is a ring of functions, for instance in $\mathcal C^\infty(\Bbb R)$. This is because multiplication is already defined to mean point-wise multiplication in such rings, and this just uses the usual relation between multiplication and exponentiation. (Convention also favours this interpretation when $f$ is a trigonometric function and $n\neq-1$, but that is really just a bad habit.)
In any other setting the more usual interpretation of $f^n$ would be $n$-fold self-composition. Because of that I think you should probably be more explicit to avoid the ambiguity, so just write
$$
  x\mapsto f(x)^n
$$
which is a couple of symbols more than $f^n$, but self-explanatory and unambiguous.
A: The notation $f^n(x)$ is ambigious. Sometimes it means $f^n(x)=\underbrace{f(f(f(\ldots)))}_{n}$, while other times, it means $f^n(x)=(f(x))^n$.
For example, $\sin^2 x=(\sin x)^2$
A: The way i read it, raising a number gives a power of the measure, so
$\sin^2 x$ corresponds to x sin square, and one is supposed to read down the tower of powers, not up them.
Multiple application, like $\sin \sin x$ = x, sin, sin, is not that common, and is best served by sequencing the terms from inner right to outer left.
