A relation is a set of ordered pairs, where the set of first  components in the ordered pair is called the domain, and the set of second components is called the range. We will typically use x to represent the first set and y to represent the second set. So ANY collection of ordered pairs (x,y) is a relation. That means a collection like {(1,2), (3,4), (5,6)} is a relation that has three items because it has three points. Another example: {(x,y)| y= 2x+1}  another relation- that is just the infinite collection of points (x,y) that lie on the line y=2x+1.

A function is a relation that is further narrowed down. It had the additional rule that for every input there must be a unique output. Roughly, that means you can’t repeat any x’s. A repeat on the x’s (with different y values) means that THAT input does not have a unique output. For example {(1,2), (1,3)} is not a function because the input x=1 does not have a unique output. It has the output y=2 and the output x=3. That’s bad = not a function. That is hard to make sense of. Watch the videos below.

Introduction to functions:

Difference between an “equation” and a “function:”

Khan Academy on functions:

Functions