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

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

Frank Vel
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  • It's not even possible to generally define $f^{1/2}$ in any particularly obvious way. – Thomas Andrews Feb 04 '15 at 13:42
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    In this context $f^{-1}$ does not describe raising the function to the power of $-1$, but rather the inverse function – Eff Feb 04 '15 at 13:42
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    Can you clarify your ambiguous notation? By $f^{1/2}$, do you mean a function $g$ such that $g(x) =\sqrt{f(x)}$ for all $x$, or a function $h$ such that $h(h(x))=f(x)$ for all $x$? The latter is a "fractional iterate", while the former is simply a power. – MPW Feb 04 '15 at 13:47
  • I didn't invent this notation, and sorry if my wording is ambiguous, I'm not entirely aware of the corresponding English phrases. I meant $f^{\frac{1}{2}}\circ f^{\frac{1}{2}} = f$ of course. Maybe $f\overset{1/2}{\circ}$ would be better? – Frank Vel Feb 04 '15 at 14:26
  • Could you provide a source where you saw this notation used to denote what you say (since you did not invent it)? – Did Feb 11 '15 at 16:12

1 Answers1

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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 $a$.

John Bentin
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