Mapsto notation

In math­e­mat­ics, the ar­row \(\mapsto\) (which in LaTeX is called “\map­sto”) is used to de­scribe what a func­tion does to an ar­bi­trary in­put. It is com­monly used in com­bi­na­tion with colon-to no­ta­tion, which de­scribes what a func­tion’s do­main and codomain are.

By it­self, \(\mapsto\) can be used to de­scribe a func­tion with­out nam­ing it, and so is a way to de­scribe anony­mous func­tions in math­e­mat­ics.


The func­tion \(f(x) = x^2\) from the real num­bers to the real num­bers can be de­scribed us­ing a com­bi­na­tion of colon-to no­ta­tion and map­sto no­ta­tion as

$$f : \mathbb{R} \to \mathbb{R}$$
$$x \mapsto x^2.$$

In one line, al­though this is less com­mon,

$$f : \mathbb{R} \ni x \mapsto x^2 \in \mathbb{R}.$$

(see in no­ta­tion).