$f(x) = \Box$ vs $x \mapsto \Box$

Question

In the book I am reading (Abstract Algebra, Dummit & Foote), the author uses 2 ways to define functions:

$$f(x) = \Box$$ $$x \mapsto \Box$$

It's not that I don't know what they mean - it's that they use both, which leaves me feeling like I am missing something, when a particular choice is used.

For example, just a few lines apart, they write a group action as

$\sigma_{g}: A \rightarrow A$ defined by $\sigma_{g}: a\mapsto g \cdot a$

and a group homomorphism as

$\varphi:G \rightarrow S_{n}$ defined by $\varphi (g) = \sigma_g$

Is there a reason one form would be used over the other?

Answer 1

Long comment ( hope it helps...)

See : Saunders Mac Lane & Garrett Birkhoff, Algebra, AMS (3rd ed., 1991), page 4 :

A function $f$ on a set $S$ to a set $T$ assigns to each element $s$ of $S$ an element $f(s) \in T$, as indicated by the notation

$s \mapsto f(s), \ \ \ \ s \in S$.

The element $f(s)$ may also be written as $fs$ or $f_s$, without parentheses; it is the value of $f$ at the argument $s$. The set $S$ is called the domain of $f$, while $T$ is the codomain. The arrow notation

$f : S \to T \ \ \text {or } \ \ S \stackrel{f}\longrightarrow T$

indicates that $f$ is a function with domain $S$ and codomain $T$. [...] We systematically use the barred arrow to go from argument to value of a function and the straight arrow $S \to T$ to go from domain to codomain.