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 each vertical line as an x-value, then intersecting the graph of a relation at exactly one point implies that a single x-value is 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 to more than one value of y. This condition causes the relation to be disqualified as a function.
So here’s the deal!
Here are some examples of relations that are also functions because they pass the vertical line test.
Cutting the Graph at Exactly One Point
Graph of the line f (x) = x + 1
Graph of the parabola f (x) = x2 − 2
Graph of the cubic function f (x) = x3
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 the Graph in More Than One Point
Graph of the “sideway” parabola x = y2
Graph of the circle x2 + y2 = 9
Graph of the relation x = y3 − y + 2