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.