# How do you know if x=15 is a function?

Dec 19, 2017

It is not a function.

#### Explanation:

A function is essentially a map between $x$ values and $y$ values. That is, for any given $x$ value there must be at most one possible $y$ value. The set of $x$ values for which there is a corresponding $y$ value is called the domain of the function.

In our example, the only $x$ value for which there are any $y$ values is $x = 15$, and then the $y$ value is totally unrestricted. For example, both $\left(15 , 0\right)$ and $\left(15 , 1\right)$ belong to the set of $\left(x , y\right)$ values satisfying the equation.

So $x = 15$ does not define a function.

Another way of describing the condition above is that $x = 15$ fails the vertical line test - the requirement that any vertical line intersects the graph in at most one point. This fails when $x = 15$ because a vertical line at that point intersects the graph at infinitely many points.