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.
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3 Answers
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Intuitively, it means that for every $x \in R$, the function f will give back a value $f(x) \in R$.
For example, a function $f(x)=1/x$ is only defined for those $x \in R$ Real Numbers $R$ that are different from $0$, so you should write $f: R/\{ 0 \} \rightarrow R$.
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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 $B$.
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It means that its domain and co-domain are real numbers. Do not confuse co-domain with range/image. The range of a function is indeed a subset of co-domain which consists of all of outputs of the function. .
