A function is a relation for which each of the value from the set the first components of the ordered pairs is related with exactly one value from the set of second components of the ordered pair.

Let's see if we can make out just what it means.  Let's take a look at the given example that will expectantly help us figure all this out.

Example:  The following relation is a function.

{(-1, 0)  (0, -3) ( 2, -3)  (3, 0)  ( 4, 5)}

Solution

From these ordered pairs we contain the following sets of first components (that means. the first number through each ordered pair) and second components (that means the second number through each ordered pair).

1^st components : {-1, 0, 2, 3, 4}                      2^nd   components : {0, -3, 0, 5}

 For the set of second components observed that the "-3" occurred in two ordered pairs however we only listed it once.

In order to see why this relation is a function just picks any value from the set of first components. After that, go back up to the relation and determine every ordered pair wherein this number is the first component & list all the second components from those ordered pairs. The list of second components will contain exactly one value.

For instance let's select 2 from the set of first components.  From the relation we see that there is accurately one ordered pair along with 2 as a first component, ( 2, -3) .  Thus the list of second components (that means the list of values from the set of second components) related with 2 is exactly one number, -3.

Notice that we don't care that -3 is the second component of second ordered par in the relation. That is completely acceptable.  We just don't desire there to be any more than one ordered pair along with 2 as a first component.

We looked at single value through the set of first components for our fast example here but the result will be the similar for all the other choices.  Regardless of the option of first components there will be accurately one second component related with it.

Thus this relation is a function.

In order to actually get a feel for what the definition of a function is telling us we have to probably also check out an instance of a relation that is not a function.