Notational differences b/w Iterated functions & exponentiation.

https://en.wikipedia.org/wiki/Exponentiation#Iterated_functions https://en.wikipedia.org/wiki/Function_composition#Functional_powers https://calculus.subwiki.org/wiki/Higher_derivative

Why does exponentiation and iterated functions have similar notation as well as names? Are these two operations similar in any way? If yes, then what are some key points of differences and similarities between the two?

I'm aware of the fact that other notations for iterated functions exist too, but that doesn't seem to reason why the common notation for exponentiation is also the same for iterated functions as well as higher order derivatives.

Solution 1:

Mathematicians like compact notation for many reasons (ease of writing and pattern recognition being the most important ones).

Unfortunately, there are not that many compact ways of combining a number and a function. So if you are in an area where you need iterated functions frequently, why would you restrict yourself from using $f^n$ just because other parts of math use that notation for exponentiation (and vice versa)?

With a bit of experience, usually, no confusion will arise. For one, if you want to iterate a function, its domain and codomain need to agree, and if you want to exponentiate, the codomain needs to have some kind of multiplication. Often, only one of these will be the case and there is only one interpretation. In all other situations, context (and good surrounding text!) is key.