Is $(f \circ g)(x) = g(f(x))$ Common in Group Theory? In the text Group Theory and the Rubik’s Cube by Janet Chen, definition 1.11. on page $4$ says

If $f:S_1 \to S_2$ and $g:S_2 \to S_3$, then we can define a new function $f\circ g: S_1 \to S_3$ as $(f \circ g)(x) = g(f(x))$ 

As far as I am aware, the order is reversed compared to what I've encountered before. How common is this in group theory, or perhaps in all of mathematics?
 A: It's not uncommon, though perhaps not common. I first learned abstract algebra through Herstein's book, which uses reversed-order composition, and later through Dummit and Foote's book, which uses ordinary order composition.
I opened my copy of Herstein to say what he says about this. I copy a relevant passage.

Let $\sigma$ be a mapping from $S$ to $T$; we often denote this by writing $\sigma: S \to T$. If $t$ is the image of $s$ under $\sigma$ we shall sometimes write this as $\sigma:s \to t$; more often, we shall represent this fact by $t = s\sigma$. Note that we write the mapping $\sigma$ on the right. There is no overall consistency in this usage; many people would write it as $t = \sigma(s)$. Algebraists often write mappings on the right; other mathematicians write them on the left. In fact, we shall not be absolutely consistent in this ourselves; when we shall want to emphasize the functional nature of $\sigma$ we may very well write $t = \sigma(s)$.

Later, with composition, he writes

We have a mapping $\sigma$ from $S$ to $T$ and another mapping $\tau$ from $T$ to $U$. Can we compound these mappings to produce a mapping from $S$ to $U$? ... Note that the order of events reads from left to right; $\sigma \circ \tau$ reads: first perform $\sigma$ and then follow it up with $\tau$. Here too the left-right business is not a uniform one.

A: It's not that common, but it's certainly done in more than one text, especially when it comes to talking about multiplication of permutations in the symmetric group.
Note that the annoying fact that $f \circ g$ means "do g, then f" is really an artifact of the decision to write "$f$ evaluated at $x$" as $f(x)$. If instead we wrote it as $(x)f$, then function composition notation would make sense. Obviously it is too late to change this.
Since we choose to write function evaluation as $f(x)$, either $f \circ g$ has to mean the functions are performed in the reverse order from what is written (standard usage) or it has to reverse symbolically when we write what it means (the usage in your source and a few others).
A: This is irregular in all the current maths literature that I know. We are usually taught to read the little circle as the verb "follows". But it need not be an error if the author is carefully consistent; just confusing.
A: I am answering myself, because I missed the comment just below this definition which makes this quite clear:

"We are using this convention because it
  matches the convention usually used for the Rubik’s cube."

Thus, unless someone adds interesting details, the answer is:
No, it is not common. However, it (supposedly) is common in Rubik's cube notation.
