The Definition(s) of a Function
Is Yours in Here?

The idea of a function is central for understanding mathematics, science, and applications of mathematics.  There are many ways to state the definition of a function.  There is no agreement on the best wording for the concept.  The important thing is to understand what a function is (have a definition that makes sense to you) and be able to relate other (which may be strange) wordings to your definition.  Here are four wordings of the definition of a function, and some comments to help you relate the definitions.  

Definition 1:	A function is a rule in which each input has exactly one output.

Comments:  This conveys the idea, it is short, and uses the terms input and output rather than x and y.

Definition 2:	A function f from a set A to a set B is a rule of correspondence that assigns to each element x in the set A exactly one element y in the set B.

Comments:  Notice the phrase changes:  "input" becomes "element x in the set A", and "output"  becomes "element y in the set B."  This definition is actually quite similar to Definition 1, but is much more formal.

Definition 3:	A function is a set of ordered pairs in which each first component is paired with exactly one second component.

Comments:  For each input there is one output.  So, make a pair out of these.  (Example:  If the input is 2, and the output is 5, then the pair is (2,5).)  These are called ordered pairs because order is very important.  Once you've formed the pairs, then put them all in one big set.  You've got yourself a function--under Definition 3.

Definition 4:	A function is a relation in which each element of the domain is paired with exactly one element of the range.

Comments:  This definition assumes you already understand the concept and terminology of relations.  Here's the translation:  "relation" = "set of ordered pairs";  "element of the domain" = "first component" = "input";  "element of the range" = "second component" = "output."  Definitions 3 and 4 are really quite similar.  Note that in Definitions 3 and 4 a function is really a "different thing" than in Definitions 1 and 2.  In the last two definitions, a function is a set of ordered pairs, and in the first two definitions a function is a rule.