How to identify the graph of a function?

Answer 1

Every function is a graph. So the only thing is to distinguish functions from other graphs.

One formal convention actually define function as its graph, and a graph is the set of all ordered pairs (x, y)

A function is a special graph where it's set set of all ordered pairs (x, y) where y = f(x). f(x) is unique (or rather one goes in only one comes out), meaning for each x, there is one and only one y. (Note: For each y, there might be many x)

So to test this, we use a "vertical line test". The idea is for all x in the domain of f, say A, we draw a vertical line (x = a for some a in A), it only intersect the graph of f one and only once. Of course, there are infinity many points, you have to do it infinitly many times. Therefore, you can do it generacally:

Let A:= dom f

For all a in A, f is a function if and only if (x = a implies f(x) = f(a) and nothing else)