TOPICS
Search

Vertical Line Test


The vertical line test is a graphical method of determining whether a curve in the plane represents the graph of a function by visually examining the number of intersections of the curve with vertical lines.

The motivation for the vertical line test is as follows: A relation f:A->B is a function precisely when each element x in A is matched to at most one value f(x) in B and, as a result, any vertical line in the plane can intersect the graph of a function at most once. Therefore, the vertical line test concludes that a curve in the plane represents the graph of a function if and only if no vertical line intersects it more than once.

A plane curve which doesn't represent the graph of a function is sometimes said to have failed the vertical line test.

VerticalLineTest

The figure above shows two curves in the plane. The leftmost curve fails the vertical line test due to the fact that the single vertical line drawn intersects the curve in two points. On the other hand, the vertical line test shows that the rightmost curve is a function on its domain: Indeed, none of the vertical lines drawn intersect the curve in more than one point and, by observation, neither would any other vertical line.


See also

Complex Function, Map, Multivalued Function, Pathological, Real Function, Single-Valued Function, Special Function

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha

References

Lial, M.; Hornsby, J.; and Schneider, D. I. Precalculus 4th Edition. Boston, MA: Pearson, 2009.Stewart, J. Calculus: Concepts and Contexts, 4th Edition. Belmont, CA: Cengage Learning via Brooks/Cole, 2010.

Cite this as:

Stover, Christopher. "Vertical Line Test." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/VerticalLineTest.html

Subject classifications