Function

A function is a relation between two sets: the domain D and the range R. The domain is the set of all possible x-values which will output the y-values. The set of possible y-values is called the range. Each input (x-value) must have only one output (y-value).

This is a function, because for each input x {1 ; 2 ; 3 ; 4} (this set is referred as domain), there is only one output y. The range of this function is {1 ; 2 ; 3}.

We can also write this function as a list of ordered pairs (x,y): {(1,2), (2,3), (3,2), (4,2)}. Or draw a graph:

What is not a function? When at least one x is associated with more than one y as shown in the picture below (1 outputs 1, 2 and 3 at the same time). The function is a relation in which each possible input value (x) leads to exactly one output value (y).

In the figure below, the pink shapes are graphs of functions. The green shapes cannot represent a function because they assign more than one y to at least one x (shown, for example, by the dashed line in point 3). A hollow dot of the pink graph means that the domain starts exactly after this point, but excludes the given boundary point, the solid dot means that the given point is included in the domain. The domain of the function with dots in the figure is therefore: