# Vertical Line Test

The **vertical line test** is a method that is used to determine whether a given relation is a function or not. The approach is rather simple. Draw a vertical line cutting through the graph of the relation, and then observe the points of intersection.

Why does this work?

The vertical line test supports the definition of a function. That is, every x**-value** of a function must be paired to a single y**-value**. If we think of a vertical line as an infinite set of x**-values**, then intersecting the graph of a relation at exactly one point by a vertical line implies that a single x**-value** is only paired to a unique value of y.

In contrary, if the vertical line intersects the graph more than once this suggests that a single x**-value** is being associated with more than one value of y. This condition causes the relation to be “disqualified” or not considered as a function.

So here’s the deal!

If a vertical line intersects the graph in all places **at exactly one point**, then the relation is a **function**.

Here are some examples of relations that are also functions because they pass the vertical line test.

## Cutting or Hitting the Graph at __Exactly One Point__

Graph of the line f\left( x \right) = x + 1

Graph of the quadratic function (parabola) f\left( x \right) = {x^2} - 2

Graph of the cubic function f\left( x \right) = {x^3}

If a vertical line intersects the graph in some places at **more than one point**, then the relation is **NOT** a function.

Here are some examples of relations that are NOT functions because they fail the vertical line test.

## Cutting or Hitting the Graph in __More Than One Point__

Graph of the “sideway” parabola *x *= *y*^{2}

Graph of the circle *x*^{2}* + **y*^{2}^{ }= 9

Graph of the relation *x *= *y*^{3}^{ }− *y* + 2

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