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A function is a relation in which each input is assigned to exactly one output. The set of all possible inputs is called the domain of the function and the set of all possible outputs is called the range. If represents the inputs and the outputs of a function, it is often said that is a function of or that depends on

This way of representing the dependent variable is called function notation. A function can be represented using a table, a mapping diagram, an equation, or a graph.
Representations of functions
Note that every function is a relation, but not every relation is a function. In the following applet, three different relations are analyzed to determine whether they are functions.
Three different mapping diagrams. The second one is not a function because one input has two outputs.
In Relation III, although one of the outputs corresponds to two different inputs, it is still a function because each input has exactly one output. Depending on how a relation is represented, there are different methods to determine whether or not it is a function.
Determining Whether a Relation Is a Function
If represented as Use
a set of coordinates or a table of values a mapping diagram
a graph in the coordinate plane the vertical line test