Is there an advantage for writing function arguments on the right side as $f(x)$ rather than on the left side as $xf$? The latter looks more natural if we think about it in diagram as $domain \xrightarrow f codomain$, and it will make the function composition easier?

  • $\begingroup$ $x(f\circ g) = (xg)f$ looks a bit awkward. $\endgroup$ – Prahlad Vaidyanathan Dec 19 '13 at 15:50
  • $\begingroup$ Some people do use that notation (and they have differnt opinions about $f\circ g$ as well). $\endgroup$ – Hagen von Eitzen Dec 19 '13 at 15:55
  • $\begingroup$ @PrahladVaidyanathan, the point of the $xf$ notation would be to interpret $f\circ g$ as "first $f$, then $g$", so that $x(f\circ g)= (xf)g$ $\endgroup$ – Mikhail Katz Dec 19 '13 at 16:05
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    $\begingroup$ Have a look at math.stackexchange.com/q/506762/75923 (for instance the comment of Ronnie Brown there). $\endgroup$ – drhab Dec 19 '13 at 16:19
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    $\begingroup$ There was sort of a push for this in the 60s, but I think it fizzled out. Lawvere pushes for this convention, for instance, in "The elementary theory of the category of sets". $\endgroup$ – Steven Gubkin Dec 19 '13 at 17:54

Putting the operator to the right is indeed how it is done in reverse polish notation. I once saw a claim in Abraham Robinson's book that the origin of the $f(x)$ notation derives from Newton. Certainly this would have to originate from late 16th or early 17th centuries.

Note: thanks to @Michael Hoppe for pointing out that the notation $f(x)$ is due to Euler in 1734.

  • $\begingroup$ $x\,\sin$ is sort of RPN, which I appreciate on my trusty HP50. However, it was Leibniz -- the grand wizard of notation -- who invents that notation. And as Voltaire put it: «Le mieux est l'ennemi du bien.» Since a quarter of a millennium no enemy was encountered. $\endgroup$ – Michael Hoppe Dec 19 '13 at 16:24
  • $\begingroup$ @MichaelHoppe, thanks for your comment. Do you have a reference for attributing the $f(x)$ notation to Leibniz? I am a big fan of the latter and would be happy to be able to attribute this to him, as well. $\endgroup$ – Mikhail Katz Dec 19 '13 at 16:27
  • $\begingroup$ Well ... no. I thought is was due to Leibniz, but it seems to be Leonard Euler who introduced $f(x)$ in 1734. I've searched eulerarchive.maa.org/date-write.html for a reference, but didn't succeed yet. Anyway, I thank you for correcting my knowledge concerning the inventor of $f(x)$. $\endgroup$ – Michael Hoppe Dec 19 '13 at 17:04
  • $\begingroup$ @MichaelHoppe: that's what I thought. Leibniz did not work with functions so much, though he is considered the one to have introduced the concept in his 1684 paper. He didn't work with derivatives either for the most part, but rather with $dx$ and $dy$. $\endgroup$ – Mikhail Katz Dec 19 '13 at 17:27

In some cases, the function s placed to the right of the variable even in modern notation, as in $n!$.

  • $\begingroup$ the derivative $f'$ too $\endgroup$ – Ooker Nov 15 '17 at 1:19

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