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One to One Function
Range, Domain, horizontal line test
A one to one function is a function in which every element in the range of the function corresponds with one
and only one element in the domain.
Example of a one-to-one function:
{ (0,1) , (5,2), (6,4) }
- Domain: 0, 5, 6
- Range: 1,2, 4
Each element in the domain (0, 5, and 6) correspond with a unique element in the range. Therefore this function is
a one-to-one function
The two functions 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 ?
Function #1 { (2,1), (4,5), (6,7), (8,9) }
Function #2 { (3,4), (8,5), (6,7), (22,4) }
Function #3 { (-3,4), (21,-5), (0,0), (8,9) }
Function #4 { (9, 19), (34,5), (6,17), (8,19) }
Answer
Relation #1 and Relation #3 are both one-to-one functions.
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Practice Problem three
Is the function below a one to-one function ?
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
X cannot be 11, 5, or 17.
If x was 17 for instance, the function would be:
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{ (8, 11), (34,5), (6,17), (12 ,17) }
In this relation, the x-value of 8 has two distinct y values and therefore this relation would NOT be a function since
each element in the domain of a function in math must have 1 and only value in the range.
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.
The Horizontal Line Test
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.
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?
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?
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
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Student Worksheet on one to one functions
