Math Homework. Do It Faster, Learn It Better.

The Vertical Line Test

You can check whether a graph represents a function by using the vertical line test.

The test states that a graph represents a function if and only if all vertical lines intersect the graph at most once.

Example :

This graph does not represent a function: for example, the vertical line

x = 1

intersects it in three places.


This graph does represent a function, since any vertical line intersects the graph exactly once, except

x = 0 ,

which doesn't intersect it at all.


Consider the graph shown.

You can see that no vertical lines pass through two points on the graph.

Note that ( 2 , 0 ) and ( 1 , 2 ) are open points and are not on the graph. So, the vertical lines x = 2 and x = 1 pass through only one point on each of the line segments.

Therefore, the graph shown is of a function.

The step function can be defined as:

y = 1 2 x + 1 y = x 1 y = 5 6 x < 2 2 x < 1 1 x 6

The domain of the function is 6 x 6 and the range is 2 y 5 .