A function is a set of ordered pairs (x,y) where for each x value there is only one corresponding y value, and the relation between x and y is such that for any x value there is a unique y value.
One way to determine if a set of ordered pairs is a function is to use the vertical line test. This test involves drawing a vertical line on the coordinate plane and seeing if it intersects the graph of the ordered pairs at more than one point. If a vertical line intersects the graph of the ordered pairs at more than one point, then the set of ordered pairs is not a function. If a vertical line intersects the graph of the ordered pairs at only one point, then the set of ordered pairs is a function.
Another way to determine if a set of ordered pairs is a function is to use the rule of correspondence. If every first element of the ordered pair corresponds to one and only one second element, then the set of ordered pairs is a function, otherwise is not.
Additionally, you can also use algebraic methods like the definition of a function, where a function is a relation from a set A to a set B, such that for each element in A there is exactly one element in B.
For example, the set of ordered pairs {(1,2), (2,4), (3,6)} is a function because for each x-value (1,2,3) there is only one corresponding y-value (2,4,6) and this is unique.
In contrast, the set of ordered pairs {(1,2), (2,4), (3,6), (1,3)} is not a function because for x = 1 there are two different y values, 2 and 3.
It's important to remember that it's not enough to just check a few ordered pairs, you need to check all of them.