I know that a function is called "function" because it's an "execution" of operations. Abbreviated notation is f. But why do we write f(x) and not (x)f or f_x or f-x- etc. ?

  • $\begingroup$ Besides the ones you list, there is also $x^\sigma$ for function $\sigma$ applied to argument $x$. $\endgroup$ – Gerald Edgar Sep 17 '16 at 21:54
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    $\begingroup$ The notation (x)f has been used. For example, Herstein's Topics in Algebra writes functions and operators on the right. At least the edition available 20 years ago did. I believe algebraists starting in the 1960s promoted this notation, perhaps due to its nice compatibility with drawing functions between sets above an arrow, but it never took off. $\endgroup$ – KCd Sep 17 '16 at 22:47
  • $\begingroup$ Polish notation dispenses with brackets altogether, en.wikipedia.org/wiki/Polish_notation $\endgroup$ – Margaret Friedland Sep 18 '16 at 16:53
  • $\begingroup$ function application is often (usually?) written without parens, e.g f x. the order, I speculate, is simply due to the left-to-right ordering of latinate writing systems. $\endgroup$ – mobileink Sep 19 '16 at 20:35
  • $\begingroup$ The linear algebra book by H. Rose (books.google.de/…) uses the notation $xf$. $\endgroup$ – student Apr 19 '17 at 21:00

See Leonhard Euler :

si $f(\frac x a + c)$ denotet functionem quamcunque ipsius $\frac x a + c$

The definition of function was already present into:

  • Johann Bernoulli, Remarques sur ce qu'on a donne jusqu'ici de solutions des problemes sur les isopdrimitres, published in Mem.Acad.roy.sci, Paris, 1718. See Opera omnia, Tomus II, page 241:

Definition. On appelle ici Fonction d'une grandeur variable, une quantité composée de quelque maniére que ce soit de cette grandeur variable et de constantes.

In the same mémoire [page 243] Bernoulli proposed the Greek letter $\phi$ as a notation for the caractéristique of a function, writing the argument without brackets: $\phi Pb$.

I presume that the improved symbolism due to Euler was devised in order to avoid the mistake of interpreting justaxposition as multiplication.

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  • $\begingroup$ If this was Eulers first use of the function notation, it's tempting to speculate that the tradition of carrying along brackets around the argument originated from this example, where brackets were necessary. (I'm assuming brackets were already used at that time to denote precedence of operations, as in $3(2+5)$?) $\endgroup$ – Michael Bächtold Nov 2 '16 at 9:33
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    $\begingroup$ @MichaelBächtold - brackets in expression was used; see e.g. Benoulli's text linked. $\endgroup$ – Mauro ALLEGRANZA Nov 2 '16 at 9:39

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