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 
is a function
precisely when each element 
is matched to exactly one value 
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 function
graph of a function on its projection onto the horizontal
axis iff no vertical line
intersects it more than once. For a prescribed domain 
, each vertical line with horizontal coordinate in 
must intersect the function
graph exactly once.
A plane curve which doesn't represent the graph of a function is sometimes said to have failed the vertical line test.
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.
