Euler is often credited with introducing the notation \$f(x)\$, & people cite the example \$f(fracxa+c)\$, where he had khổng lồ use parentheses around the function argument. On the other hand, when the argument was a single letter like \$x\$, I have mainly seen Johann Bernoulli and Euler just write \$f, x\$ or \$fcolon x\$ (or \$phi, x\$), without the parentheses. If I recall correctly even Lagrange in his lectures introduced the function notation without parentheses.

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Question: Did Euler (or Johann Bernoulli) ever write \$f(x)\$?

In case the answer is no, the follow up question is: when did it become standard lớn put parentheses around \$x\$?

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edited Oct 7, 2021 at 13:05

Rodrigo de Azevedo
asked Jan 8, 2018 at 10:32

Michael BächtoldMichael Bächtold
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I’m guessing no. But how does one make sure? (Maybe with 85+ volumes of clean pdfs...)

Cajori, who started that \$f(frac xa+c)\$ example, points out a \$varphi(z)\$ in D’Alembert (1754, p 50).

For “standard”, I would say Lacroix (1797, p. 87):

4. Pour représenter une fonction sans indiquer, en aucune manière bình luận elle peut être composée, je me servirai de la caractéristique \$mathrm f\$; et il faudra entendre, par l"expression \$mathrm f(x)\$, une fonction quelconque de \$x\$, en comprenant sous cette dénomination tout ce que comporte la définition du mot fonction (Intr. nº 1) : on doit donc bien se garder de prendre la lettre \$mathrm f\$ pour un coefficient de \$x\$. J’indiquerai la substitution de \$x+k\$ aulieu de \$x\$ dans \$mathrm f(x)\$, en écrivant \$mathrm f(x+k)\$, et cela voudra dire que le résultat est composé en \$x+k\$, comme la fonction primitive l’est en \$x\$.

Side remark tying into your other question: This book of Lacroix writes “the function \$f\$” very often; e.g. Pp. 93, 212, 258, 483–496, 502, mainly when describing results of Monge who also did this a lot (but avoided unnecessary parentheses). I think “\$f\$” all started with solutions of PDEs depending on “arbitrary functions” — though only Dedekind, I would say, made them “objects” in the sense you want at the other question.

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Edit:In E213 “Remarques sur les mémoires précedens de M. Bernoulli” (1755), just quoted elsewhere, you can see Euler “forget” his evaluation colon and slip into writing \$Phi"(x)\$ (p. 215) and eventually \$Phi(x)\$ (p. 216). Same thing in E441 (1773, p 429). So in the end, yes.