What is a function?

Recall that a binary relation on sets of objects  and  is a collection of ordered pairs , where  is one of the objects in  and  is one of the objects in . A function is a binary relation with the additional property that each object in  occurs as the first component of at most one pair in the relation.

The domain of a function is the set of all values which appear as the first component in the function. The range of a function is the set of all values which appear as the second component in the function.

    A note on notation 
Functions are fundamental objects in mathematics, and there is a variety of notations in common use for representing and describing functions. For example, the squaring function given by the ordered pairs , where x is any number, could also be written as:

The last notation shown is very common, and arises from the standard convention of labeling the vertical axis of a graph with the letter y (and the horizontal axis with x).