Is there a graphic visualisation of $f^{\frac{1}{2}}$?

Question

$f^{-1}$ can be visualized by flipping the graph of $f$ around the line $y=x$.

Is there a similar way to describe graphs of functions to a fractional power, like that one of $f^{\frac{1}{2}}$?

Answer 1

For a limited class of functions, at least, the idea of "$f^\frac12$" does make sense. Thus, let $f(x)=|x|^a$, where $a>0$. Then you could take $f^\frac12(x)=|x|^\sqrt a$. However, there is no simple way to relate the graph of $f^\frac12$ to that of $f$. For a start, any relationship would vary with the value of $a$.