Why does the vertical line test work?

1 Answer
Jul 31, 2018


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Please read the explanation.


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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:

enter image source here

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#.

enter image source here

This mathematical relation

#color(blue)(y^2=(-4x)# is NOT a function,

as the vertical line test fails.

Hope it helps.