Vertical line test is a test to determine whether the given relation is a function or not, when the domain and range of the given problem(relation) correspond to x-axis and y=axis.
If we have a graph of a set of ordered pairs so we can easily draw a vertical line on it. If, the vertical line cuts the graph or intersects the graph at one point then the given graph is a function and if, it cuts the graph more than one point then the given graph is a relation.

## Vertical Line Test Examples

Example 1:
y = x + 1, x = 1,2,3,4 is a function or relation.
Solution:
We have S = {(1,2), (2,3), (3,4), (4,5),(5,6)}, lets plot a graph of the given set

It is clear that if we draw a vertical line(pink) then it cuts the graph at one point.Hence the given equation is a function.

Example 2:
If x2 + y2 = 4 be the given equation of circle. Determine by using the graph the circle is a function or relation.
Solution:
We have x2 + y2 = 4 is the equation of circle whose center is origin and radius is 2.
Then the graph of this is
It is clear that the vertical lines cuts the graph in ti two points, hence the given graph is a relation.

Example 3:
Determine using the equation y = $\sqrt{9 - x^{2}}$ is a relation or function.
Solution:
Given y = $\sqrt{9 - x^{2}}$ is an equation of semi circle and if we draw a graph of it this then we get,
Vertical line test shows that the vertical line cuts the graph at one point, hence semi circle is a function.

Example 4:
Does graph of the equation y = x3 +1 pass the vertical line test.
Solution:
Given equation is y = x3 +1, then graph of this
Hence the given equation is a function.

Example 5:
Does the relation x2 = 4y(parabola) is a function.
Solution:
Lets plot a graph of x2 = 4y, then
It is clear from the above graph that the given relation is a function.