What does $f|_A$ mean? If $f$ a is a function and $A$ is a set, what could the notation
$$f|_A$$
mean? Is it perhaps "restricted to set $A$"?

Intuitively speaking, a function $f$ is constituted of three ingredients:


*

*a domain;

*a codomain;

*a rule (that, for each element in the domain, assigns a unique element in the codomain).


If we change any of these three ingredients, we obtain a different function. In particular, if we change the domain by a subset $A$ of the original domain (keeping the codomain and the rule), we get a new function which is represented by $f|_A$.
In other words: given a function $f:X\to Y$ and a set $A\subset X$, the notation $f|_A$ denotes the function $g:A\to Y$ given by
$$g(x)=f(x),\quad \forall \ x\in A.$$
This is the usual meaning but, maybe, there are different meanings in other contexts.
