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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) | |c|c|c|c|c|c|c|c| x & -2.25 & -1.75 & -1 & -1 & 0 & 1 & 2 y & -2 & 0 & 2 & -1 & 1 & -0.75 & 0.5 |
To determine whether the relations are functions, follow these two steps.
Notice that l_3 cuts the first graph at two different points. The line m_2 also passes through two different points. This means that neither Relation I nor Relation III is a function. However, all of the vertical lines drawn over Relation II only intersect the graph one time at most. Because of this, Relation II is a function.
Keep in mind that before stating whether 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.
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 determining whether a relation is or is not a function, it must be assumed that the graph of a relation continues without any significant change beyond the boundaries of the coordinate plane. If this were not the case, it could never be determined from a graph whether a relation is a function.