# How do I determine whether a given relation is a function?

Nov 3, 2015

Here are a couple of methods...

#### Explanation:

Set of points

If the relation is expressed in terms of a small list of points, first eliminate any duplicates (that is equal $x$ and $y$ coordinates), then check to see if the $x$ coordinates of the remaining points are distinct. If they are distinct, the relation represents a function. If not, then not.

Equation

Try to express $y$ in terms of $x$ and see if you get a well-defined unique value for $y$ wherever defined. If you do, then you have a function, if not, then not.

For example, ${x}^{2} + {y}^{2} = 25$

Then $y = \pm \sqrt{25 - {x}^{2}}$ is not uniquely defined, so the relation fails the 'vertical line test' and is not a function.