Let us say we have a graph drawn for a mathematical relation.
This is a mathematical relation, as for every value of #color(blue)(x#, there is a corresponding value of #color(blue)(y#.
To check whether this graph is a function, a Vertical Line Test can be used.
Let us examine the following graph drawn for the mathematical relation:
Can you observe the vertical lines drawn for various values of #color(red)(x# ?
All of these vertical lines intersect the graph at one point only.
If there is just one point of intersection,
that would mean,
for every x-value, there is just one unique y-value.
This property proves that the given mathematical relation
is a mathematical function.
Hence, the vertical line test works in determining whether a given relation is a function.
Let us try one more mathematical relation:
This mathematical relation creates a parabola and the parabola is symmetric about the x-axis, at #color(red)(y=0#.
This mathematical relation
#color(blue)(y^2=(-4x)# is NOT a function,
as the vertical line test fails.
Hope it helps.