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Let $f : X \to Y$ and $g : Y \to Z$ be functions. We define the composition $g \circ f : X \to Z$ by $g \circ f(x) = g(f(x))$ for each $x \in X$.

I have also heard the composition read out like this:

  • "The composition of $f$ with $g$ is . . ."
  • "Consider the function $g$ composed with $f$, given by . . ."
  • "The function $g$ of $f$ is . . ."

Is this an appropriate way to speak of $g \circ f$? It sometimes happens that I (or my teachers) reverse the order of $f$ and $g$ when describing $g \circ f$ in any of the above ways. Surely, it can't be that both ways are correct. It doesn't cause confusion because the function being talked about is quite straightforward. But I'm still interested in knowing what the "correct" way/s to describe $g \circ f$ is/are among the above.

I know that some people prefer the functional notation that operates the other way, but this question is not about that scenario.

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  • $\begingroup$ "The composition of $f$ with $g$" is ambiguous, so it should be avoided. The other two ways to say it are fine. $\endgroup$ – JavaMan Jul 24 '18 at 14:24
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    $\begingroup$ I've heard $g\circ f$ been read as "$g$ after $f$." In any case, the $g$ should be pronounced first. Also, see this wiki article. It gives multiple pronunciations. $\endgroup$ – Crosby Jul 24 '18 at 14:34
  • $\begingroup$ @Crosby oh! That does make it unambiguous. Interesting, thanks. $\endgroup$ – Brahadeesh Jul 24 '18 at 14:36
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    $\begingroup$ @Crosby I suggest my students to read $g\circ f$ as “g dopo f” (“dopo” is Italian for “after”). I believe that this helps into remembering which function should act first. $\endgroup$ – egreg Jul 24 '18 at 14:37
  • $\begingroup$ The OP accept my answer which was down-votes, by users without the question. Got to love matchstack :). $\endgroup$ – Faraad Armwood Jul 24 '18 at 14:52
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Here's your answer via Wikipedia:

For instance, the functions $f : X \to Y$ and $g : Y \to Z$ can be composed. . . The resulting composite function is denoted $g \circ f: X \to Z$, defined by $(g \circ f)(x) = g(f(x))$ for all $x$ in $X$. The notation $g \circ f$ is read as "$g$ circle $f$", "$g$ round $f$", "$g$ about $f$", "$g$ composed with $f$", "$g$ after $f$", "$g$ following $f$", "$g$ of $f$", or "$g$ on $f$".

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  • $\begingroup$ I've made an edit. $\endgroup$ – Faraad Armwood Jul 24 '18 at 14:37
  • $\begingroup$ What a goof, I forgot to check wikipedia! Thank you, Faraad :) $\endgroup$ – Brahadeesh Jul 24 '18 at 14:41
  • $\begingroup$ No problem. I'm sorry I couldn't edit in time enough for the down-voters. Matchstack is weird like this, so make sure to be on your P's and Q's when using this site. More than mathematics has manifested itself here. Ego's will be a big issue to overcome. $\endgroup$ – Faraad Armwood Jul 24 '18 at 14:53
  • $\begingroup$ Hey, reputation points are just for fun anyway. :) You taught me a valuable lesson, I'll be sure to check Wikipedia for these sort of things before rushing to MSE, next time. $\endgroup$ – Brahadeesh Jul 24 '18 at 14:55
  • $\begingroup$ They're other point systems that users abide by :) . And yes, wiki is great for "summed" mathematics i.e they'll give you they story, but it'll be condensed like no other. $\endgroup$ – Faraad Armwood Jul 24 '18 at 14:57
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(I have not yet learnt the nitty gritty definitions of functions so I'm not sure how useful my answer is.)

When saying aloud $g \circ f$ I've always said "$g$ of $f$" - the same way you would say $g(x)$ as '$g$ as a function of $x$'. My mathematics teachers in high school and at university have exclusively said $g$ of $f$.

Your second and third examples make perfect sense to me. I find the first one confusing. The answers in the comments e.g. $g$ after $f$ also make sense.

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