what is$ f^{(n+1)}(x)$ as a function

Question

Define $f^{(n+1)}(x)$ in function form. Is it $f(f^{(n)}(x))$ or is it $f^{(n)}(x)*f(x)$. Or is it something else completey. Thank you so much. I'm actually studying functions and this was something that just poped into my head. So please let me know.

Answer 1

In terms of functional iterates, we generally write $$f^{n+1}(x)=f^n(f(x))$$ or, equivalently $$f^{n+1}(x)=f(f^{n}(x))$$ or even $$f^{a+b}(x)=f^{a}(f^{b}(x)).$$

Typically, parenthesis aren't used in the exponent of functional iterates (though I have seen it), because parenthesis are usually used to disambiguate between functional iteration (no parenthesis) and higher derivatives (with parenthesis).

Answer 2

Mathematicians are indeed not all too consistent here.

It is generally agreed that $f^{(n)}$ denotes the $n^{\rm th}$ derivative of $f$. E.g., if $f(x):=e^{\lambda x}$, then $f^{(n)}(x)=\lambda^n \>e^{\lambda x}$ for all integers $n\geq0$.

With $f^n$, $\>n\in{\mathbb Z}$, it is another matter. While everyone would interpret $\sin^2 x$ as $\bigl(\sin x\bigr)^2$ the term $f^{-1}$, as in $\sin^{-1}x$, can mean the inverse function of $f$, in the example: $\arcsin$, but it could also denote the reciprocal ${1\over f}\,$. In most cases it is clear from the context what is meant. If you want to emphasize the intended meaning you can write $f^{\circ n}$ for the $n^{\rm th}$ iterate of $f$, which is certainly preferable to $f^{(n)}$.