Composition of functions, notation and order confusion.

Question

I thought composition of functions was something that I learned long ago and had down.

However, recently I've been terribly confused by the following.

We are dealing with permutations and my professor writes (for example):

$1 \xrightarrow{\sigma} 2 \xrightarrow{\tau} 8$, and calls this $\sigma \circ \tau$.
$\therefore (\sigma \circ \tau)(1) = 8$.

This can also be written as $\tau (\sigma(x))$. This makes sense to me following the arrow notation, but initially I tried:

$\sigma(\tau(x))$ which gave the incorrect results, this is because I have historically been taught that: $(f \circ g)(x) = f(g(x))$.

For example: $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) =(x+1)^2$.

Where is the confusion coming in?

Thanks

Answer 1

The notation of your professor is indeed inconsistent as $\sigma\circ\tau$ usually should mean $x\mapsto\sigma(\tau(x))$.

However, specifically for permutations one may prefer the dual operation that reads the composition from left to right, but to distinguish it's usually denoted by some other symbol like $\sigma;\tau:=\tau\circ\sigma$ or $\sigma\cdot\tau$ or simply $\sigma\tau$.

Answer 2

There is a mistake in the two Greek letters above the arrows. From 1 to 2 it should be tau, from 2 to 8 it should be sigma. This mistake is the source of the confusion here.