What is the difference between a function and a relation

That is like saying what is the difference between a flower and a rose.

Well, a rose is a flower but a special kind of a flower.  A flower can be a rose but id doesn't have to be.

OR

What is the difference between a square and a rectangle.

Well a square IS a rectangle it is just a special type of rectangle.

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A Function is a relation but it is a special type of relation.

I know that students are often asked "Is it a function or a relation?" This wording is extremely poor. 

A relation is just a set of x and y values that are related to one another in some way.

Now a function is a special type of relation where every x value has at most one UNIQUE y value that corresponds to it.

So you put x into the forumula.  It either makes no sense and there in no real y value or there is exactly one y value.

If you draw the graph then functions pass the "Vertical line test"    This means that if you draw a vertical line anywhere on the number plane it will pass through the graph at most once.

Look at this for more discussion.

https://www.khanacademy.org/math/algebra2/functions_and_graphs/function-introduction/v/relations-and-functions

 

A