The Vertical Line Test is considered to be a mathematical method of determining whether a curve in the plane indicates the graph of a function or not. In this article, I will be discussing the Vertical Line Test and all other related things about it.
Keep reading till the end of the article to find out more information about the Vertical Line test and more!
About Vertical Line Test
Right onto answering your query on what the Vertical Line Test is exactly – when it comes to Maths, the Vertical Line Test refers to the visual way of determining whether a curve is a graph of function or not. A function can only have a single output which is “y” for each of the unique inputs, i.e “x”.
If a vertical line happens to intersect a curve on a plane that is xy-, more than time. Then what happens is that for a single value of “x,” the curve gets more than one value for “y”. Hence, the curve does not represent a function.
If all of the vertical lines happen to intersect a curve at the most – once, then the curve indicates a function.
Formula
The formula of the Vertical Line Test also includes the substitution of X=A in Y=F(X), i.e. if it happens to give a single unique value of Y=F(A). Only then will it be considered as a function. Else, if it gives more than a single value then it does not represent any function.
Keep reading just a bit more to get to know about Vertical Line Test examples!
How To Apply A Vertical Line Test?
Below, I will be listing the sequence of steps that you need to follow for applying the Vertical Line Test, to find whether a given expression is even a function or not.
There are two possible methods on how you can apply the Vertical Line Test! It can be applied either algebraically or geometrically.
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Example:
The following example is an algebraic problem which will make you better understand the concept of Vertical Line Test.
Q. Consider the function Y=F(X) and the Vertical Line has the equation of X=A.
Algebraically
Now to further clarify how the Vertical Line Test function works in Algebra. The equation of a Vertical Line “X=A” and if that gets substituted in the equation of a curve Y=F(X), then you are going to get Y=F(A).
If you happen to get more than a value for Y, then it would prove that the equation of Y=F(X) does not represent any function.
Moreover, if you get only a single value for Y on substitution of X=A in the Y=F(X), then the Y=F(X) is going to represent a function.
Geometrically
First, you need to draw the graph of Y=F(X), concerning the coordinate axis. Now, you need to draw the line X+A, and then observe the no. of places it cuts the Y=F(X) curve.
If the Vertical Line happens to cut at more than a single place then the curve does not represent any function.
Otherwise, if the curve cuts at only a single point which is the (A, F(A)), then the Y=F(X) curve represents a function.
Keep reading till the end of the article to find out even more information about the same!
Other Common Queries
Here is a list of some of the more popular questions that get asked about the topic of “Vertical Line Test”:
Q. What Can Be Derived From The Relation Of A Vertical Line Test?
A: The relation states the values of both X and Y or even the codomain/domain values with the equation of Y=F(X).
If the vertical line happens to cut through the graph of relation Y=F(X) at a distinct point then it is considered to be a function. If the curve of the graph is cut more than a single point then that would prove it is not a function.
Q. What Is The Difference Between A Horizontal Line Test And Vertical Line Test?
A: The Vertical Line Test aids in finding a given curve and whether it represents a function or not. If the Vertical Line has cut the graph of the equation at a single point then it represents a function. Else, if it cuts through more than once then it is not considered as a function.
The Horizontal Line Test, on the other hand, aids in finding whether a function is injective or not. If a Horizontal line happens to cut through the graph once, then it is considered to be “injective” in nature. Also, the function would be considered a unique inverse.
If the horizontal line has cut through more than a single point then it is considered to have the same codomain values for the various domain values, and also the function would not be considered as injective.
To Wrap It Up!
Thus, it is understood that the Vertical Line Test is very crucial as it aids a person in easily finding whether the curve represents a function or not. Through the medium of Visual Representation, one can easily find whether a curve is a function.
If the Vertical Line cuts through the curve at a single point then it is indeed a function. Or if it happens so that the curve cuts at more than a single point then it does not represent a function.
Thank you for reading up till here. I hope you found the information about the same useful.
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