Functions:Definition

From Department of Mathematics at UTSA
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Example of an arrow diagram of a function

A function (or "mapping") is a relationship between two sets and that maps each input to exactly one output . A function that maps elements of the set to elements in the set is denoted as , where is the domain of and is the codomain. We can also think of a function as a set of ordered pairs , and , such that each element is paired with exactly one element . If a function maps an input to an output , we can write that . For finite, reasonably small sets, we can depict a function graphically (see image).

Note: a function cannot map one input to more than one output, but it can map more than one input to the same output. For example, let and , and . If is a relation such that and , then is NOT a function. However, a relation such that and IS a valid function.


Examples:

  • Let and , and let such that , , , , and . Since each element of the domain maps to exactly one element (that is, there is no and such that ), is a function.
  • For sets and in the previous example, let be a relation such that , , , , and . Since maps the input to two distinct outputs, this relation is NOT a valid function.
  • Let such that . This is a function, since each maps to exactly one element .
  • Let such that . This is not a valid function, since for , can equal both and , and for .

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