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!
from Hot Weekly Questions - Mathematics Stack Exchange
cmarangu
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