I'm having trouble understanding this. My friend said it doesn't, but I disagree, though I'm not sure.
Given a function $f:A \to B$, the graph of $f$ is defined by $G(f) = \{(x,y)| x \in A , y = f(x)\}$. Then, is it true that if a function exists, its graph exist (Even though there may be no geometric interpretation)? I think this is true, since in the "worst" case it would be the empty set, which exists...
On
Actually, any relation from a set $A$ a set $B$ has a graph. Such a relation can be defined as a triple $(G,A,B)$ such that $G\subset A\times B$, and $G$ is the graph of the relation.
A function from $A$ to $B$ is just a relation $(G,A,B)$ such that, if $(x,y)$ and $(x,y')\in G$, then $\;y=y'$ (such a graph is called a functional graph).
A map from $A$ to $B$ is a function such that for each $x\in A$, there exists a (necessarily unique) $y$ such that $(x,y)\in G$.
You are right, it exists.
If you are seeking to plot a function between 2 sets, you can plot $A$ on one axis, and $B$ on the other one and pick off the points as you mentioned.
In addition, if $A$ and $B$ are finite, you can draw $A$ and $B$ as points in space, and connect them via edges. You get $V = A \cup B$ with edges $E = \{(a,b) | f(a)=b\}$, which is also a representation with a graph, albeit quite a different one from the one you are describing.