why is it called $f^{-1}(x)$? Why do we name the inverse function $f^{-1}(x)$? Is it nonstandard to say $f^{0}(x)=x$, $f^{1}(x)=f(x)$, $f^{2}(x)=f(f(x))$, $f^{\infty{}}(x)=f(f(f(⋯f(x)⋯)))$?
Some (all) functions can have a $f^{1/2}(x)$?
Like if $f(x)=x+1$ then $f^{1/2}(x)=x+1/2$ so $f^{1/2}(f^{1/2}(x))=(x+1/2)+1/2=x+1=f(x)$
Or if $f(x)=x^4$ then $f^{1/2}(x)=x^2$ so $f^{1/2}(f^{1/2}(x))=(x^2)^2=x^4=f(x)$
What about superscripts with complex numbers, which are supposed to be a natural phenomenon appearing anywhere that the negative reals can?
$f(f(x))$ adds the superscripts, what about multiplying them? This notation could be formalized with recursion. It could help describe the mandelbrot set more elegantly as $f^{\infty}(0)$ where $f^n(z)=f(n-1)^2+z$ . It could help people realise there exist smooth transitions between functions and themselves called on  themselves: $f^{1.5}(x)$
Most importantly, who invented this syntax? Is there an area of mathematics formally exploring this? Thanks!
 A: I have no idea who invented this syntax. However, I typically use $f^{\circ n}(x)$ rather than $f^n(x)$ to denote repeated composition, since $\circ$ is the symbol used to denote function composition, and $f^n$ can be easily confused with exponentiation. (Especially when using trig functions, since $\sin^2 x$ is often used to denote $(\sin x)^2$.)

Regarding why $f^{-1}$ denotes the inverse function: notice that repeated function composition satisfies the following nice addition property:
$$f^{\circ m}\circ f^{\circ n}=f^{\circ (m+n)}$$
If we want to make $f^{\circ n}$ well-defined for negative $n$, but still want this nice addition property to hold true, then we need
$$f^{\circ -1}\circ f^{\circ 1}=f^{\circ 1}\circ f^{\circ -1}=f^{\circ 0}$$
and the inverse function $f^{-1}$ satisfies the desired property.

If you’re looking for a multiplicative analogue of the additive law
$$f^{\circ m}\circ f^{\circ n}=f^{\circ (m+n)}$$
then the following might be what you’re looking for:
$$(f^{\circ m})^{\circ n}=f^{\circ mn}$$

Be careful referring to $f^{\circ 1/2}$ as if it were a unique function. For example, the function $f(x)=2x+1$ has at least two half-iterates:
$$f^{\circ 1/2}(x) =^? \sqrt{2}x+\frac{1}{1+\sqrt{2}}$$
and
$$f^{\circ 1/2}(x) =^? -\sqrt{2}x+\frac{1}{1-\sqrt{2}}$$
So again, be careful! It’s not always uniquely defined.
Regarding which functions have half-iterates: lots of them, but if you impose the restriction that the half-iterate must be continuous, it turns out that continuous decreasing functions cannot have continuous half-iterates.

Related blog posts:

*

*Fractional iterates

*A summary of functional iteration

*N-involutory rational functions

*Iterated polynomials
