Method

Vertical Line Test

The vertical line test is a graphical method to determine whether a given relation is a function. For example, consider the following relations.

Relation I	Relation II	Relation III
$(y - 2) (y + 1) = x$	$y = x (x + 1) (x - 2)$	$x y - 2.25 - 2 - 1.75 0 - 1 2 - 1 - 1 01 1 - 0.75 2 0.5$

To determine whether the above relations are functions, these two steps can be followed.

1

Draw the Relation on the Coordinate Plane

Draw the relation on the coordinate plane. The first two relations can be drawn using a graphing calculator or another mathematical software.

Graph of the three relations: (y-2)(y+1)=x; y=x(x+1)(x-2), and x (-2.25,-2),(-1.75,0),(-1,2),(-1,-1),(0,1),(1,-0.75), and (2,0.5)

2

Draw a Vertical Line and Look at the Intersection Points

Draw vertical lines at different places of the coordinate plane. If one line intersects the graph more than once, the relation is not a function. On the contrary, if no vertical line cuts the graph more than once, the relation is a function.

Different vertical lines drawn along each relation

Notice that $ℓ_{3}$ cuts the first graph at two different points. Also, $m_{2}$ passes through two different points. Then, neither Relation I nor Relation III is a function. In contrast, any vertical line cuts the graph of Relation II at most once. Thus, Relation II is a function.

Relation I is not a function. $\times$
Relation II is a function. $✓$
Relation III is not a function. $\times$

Keep in mind that, before stating that a relation is a function, the vertical lines drawn have to cover the entire domain to ensure that no vertical line cuts the graph more than once.

Why

Intuition Behind the Method

When a relation is not a function, it is because there are multiple $y -$ values that proceed from the same $x -$ value. Therefore, the graph of such relation would show at least two points directly above the other.

Points with the same $x -$ value belong to the same vertical line.

This is why drawing a vertical line and moving it across the graph reveals if the graph is a function or not.

Note that, when concluded that a graph is a function, it must be assumed that the graph of the relation continues without any significant change beyond the boundaries of the coordinate plane. If not, it could never be concluded from a graph that a relation is a function.

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