Is (5, -2), (5, -1), (5, 0), (5, 1) a function?
A function is an expression in terms of a variable, usually x. Hence, that is not a function.
What you have there are four sets of coordinates. They may have all come from the same function, but they are not themselves a function as there is no x term involved.
Function notation is essentially the same as graphical notation, but instead of 'y =' we write 'f(x) ='. This is more useful as it allows transformations applied to the function (or graph) to be easily seen, and can also be used out of the context of graph. A function is shown below to illustrate this.