One to One Function

The two functions below only differ by 1 number. However, that small difference is all that was necessary to make function #1 not be a one to one function.

In the first function below, since the number 5 in the range corresponds with both 4 and 11 in the domain, this function is not one-to-one.

On the other hand, function #2 is a one to one function because each element in the domain has 1 and only 1 corresponding element in the range.

Practice Problem Two
Which functions below are one to one ?

Function #1 { (2,27), (3,28), (4,29), (5,30) }
Function #2 { (11,14), (12,14) , (16,7), (18,13) }
Function #3 { (3,12), (4,13), (6,14), (8,1) }

    Answer   
Relation #1 and Relation #3 are both one-to-one functions.

Practice Problem Two

Which functions below are one to one ?

Practice Problem three

Is the function below a one to-one function ?


Answer

Yes, because every element in the range is matched with only 1 element in the domain.

Practice Problem Four

For the following function to be one-to-one, X can not be what values?


{ (8, 11), (34,5), (6,17), (12 ,X) }

    Answer    

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Practice Problem Four

For the following function to be one-to-one, X can not be what values?


{ (21, 22), (22,15), (111,113), (12 ,X) }

    Answer    
X cannot be 22, 15, or 113.

If a function is one to one, then the function not only passes the vertical line test, but it also passes the horizontal line test.

The Horizontal Line Test : If a horizontal line only intersects with the graph of a function once, then this function is one-to-one. If a horizontal line intersects the graph of the function more than once, then this function is not one to one.

At the top of this page we examined at function #1 and function #2 below.
Let's examine what function #1 looks like in a graph.

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Since the horizontal line intersects the graph of the function below, this function is not one-to-one.

Is the function in the graph below one to one?
    Answer    
No matter where we draw it, a horizontal line only intersects the graph once. Therefore, this is a one-to-one function.

Is the function pictured in the graph below a one-to-one function?

Answer

The graph does not represent a one to one function. Although this graph passes the vertical line test. The graph does not pass the horizontal line test.

Is the function pictured in the graph below a one-to-one function?

Answer

The graph does not represent a one to one function. In fact, it is not even a function because this graph does not pass the vertical. There is no need to do the horizontal horizontal line test if a graph does not pass the vertical line test. Remember that the horizontal line test is simply used to distinguish a special case of a function (that of 1 to 1 function). If it doesn't pass the vertical, it's not even a function to begin with.

Is the function below one to one?
    Answer    
No matter where we draw it, a horizontal line only intersects the graph once. Therefore, this is a one-to-one function.

Are all lines one to one functions?
    Answer    
No,

Vertical lines such as x= 2 are not functions at all.
Horizontal lines such as y = 9 are functions but they are not one to one functions.

All other lines are indeed one to one functions