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

a vertical line intersecting a line at exactly one point

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

a vertical line intersecting a quadratic function at exactly one point

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

a vertical line intersecting a cubic function at exactly one point

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}

a vertical line intersecting a sideway parabola at two points

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

a vertical line intersecting a circle at two points

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

a vertical line intersection a graph at three points

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Horizontal Line Test