Many ways to give a function

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"There are many ways to give a function: by a formula, by a plot or graph, by an algorithm that computes it, or by a description of its properties. Sometimes, a function is described through its relationship to other functions (see, for example, inverse function). In applied disciplines, functions are frequently specified by their tables of values or by a formula. Not all types of description can be given for every possible function, and one must make a firm distinction between the function itself and multiple ways of presenting or visualizing it."

This is a quote from: http://en.wikipedia.org/wiki/Function_(mathematics [Broken])

in the last sentence he mentions "one must make a firm distinction between the function itself and multiple ways of presenting or visualizing it."

I`m now confused because when I think of a function I think of one of its representations e.g. formulas; the author says that I should make a distinction between the ways of presenting/visualizing and the function itself.. how do I do so? and when the word "function" is mentioned how should I interpret it in my mind?
 
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Well you can think of a function any way you want really as long as you understand what it's defined to be. A f function is a set of ordered pairs such that if f(x) = y and f(x) = y' (or (x,y) and (x,y') are members of the set) then y = y'. It's useful to think about it as a curve in a plane or a set of ordered pairs or to look at an equation that can be used to find the values of the function depending on the context. I think that the Wikipedia page wanted to make the point that not all functions can be represented in each form; so if a function is defined in an way which you are unfamiliar with, and not as an equation or graph, then that doesn't mean that the function isn't defined properly or isn't a genuine function etc.
 

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