# Why does the vertical line test work?

Jul 31, 2018

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#### Explanation:

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Let us say we have a graph drawn for a mathematical relation.

color(red)(y=f(x)=x^2

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

color(red)(y=f(x)=x^2

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:

color(blue)(y^2=-4x

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.