Difference between a function and a graph of a function? Formally, I learned that a function $f: X \to Y$ is a subset $f \subset X \times Y$ subject to the condition that for every $x \in X$, there is exactly one $y \in Y$ such that $(x, y) \in f$. We write $y = f(x)$.
The graph of a function $f: X \to Y$ is the set $\{(x, f(x)) : x \in X\}$.
Based on these two definitions, the graph of a function is, well, just the function (i.e $\operatorname{graph} f = f$), which makes me wonder why would you consider the "graph" of a function if it doesn't tell you anything new?  

What’s going on here is that “everybody knows” what a function ought to be, and the definition of a function as being equal to its graph is just a tool for codifying that. I assure you that there’s hardly a working mathematician in the world who thinks of a function as equaling its graph. A perfectly analogous situation is the standard definition of what an ordered pair $(a,b)$ “is”. (Something like $(a,b)=\{\{a\},\{a,b\}\}$, or another equally unintuitive construct.) Giving the “definition” is just giving a set-theoretic construction that is logically sound and delivers the properties that everybody was expecting the concept to satisfy.
