What is the correct way to read $f\circ g$?

Question

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.

Answer 1

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 $f$".

Answer 2

(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.