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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
(y2)(y+1)=x y=x(x+1)(x2)
To determine whether the above relations are functions, these two steps can be followed.
1
Draw the Relation on the Coordinate Plane
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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
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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 cuts the first graph at two different points. Also, m2 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.
  • Relation II is a function.
  • Relation III is not a function.

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.

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Why

Intuition Behind the Method

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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.
Moving a vertical line across two different graphs
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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