TITLE: What does $R \rightarrow R$ means in functions?
QUESTION [6 upvotes]: I have a function.
The function is:
$$ f:R \rightarrow R $$
$$ f(x) = x^3$$
What does $R \rightarrow R$ means?
I don't know what types of questions should I ask here. If it is not ok, the I will delete.

REPLY [3 votes]: Actually a function is a subset of a product $A\times B$ that has the property that for every $a\in A$ there exists exactly one $b\in B$ such that pair $\langle a,b\rangle$ belongs to it. Then $\langle a,b\rangle\in f$ wich means the same as $f(a)=b$. 
In that context the notation $f:A\rightarrow B$ is used. The set $A$ is by definition the domain of the function and the set $B$ is the codomain. It is also called "range" instead of codomain, but often authors make a distinction between range and codomain and the range is then defined as $\{b\in B\mid\exists a\in A [b=f(a)]\}$ wich is a subset of $B$.