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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.
  • Picture of one to one function
    Practice Problems
    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   


    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    
    `
    Practice Problem three
      Is the function below a one to-one function ?
      one to one function
      Answer

    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    
    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    

    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.
    At the top of this page we examined at function #1 and function #2 below. Picture of one to one function
    Let's examine what function #1 looks like in a graph.
    Since the horizontal line intersects the graph of the function below, this function is not one-to-one.
    Picture of the horizontal line test
    Is the function in the graph below one to one?
        Answer    
    Is the function pictured in the graph below a one-to-one function?
    Answer

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

    Is the function below one to one?
        Answer    
    One To One Function Problem
    Are all lines one to one functions?
        Answer    


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