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What is an one-to-one function?

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One to one function is a function in which for every X, in domain,there is one and only one value for f( X)
 
  i.e.   a linear function: 
      
          f( X) = 2X + 3
 
        X= 2   f(x) = 2* 2 + 3 = 7 , it can be tried for all X
 
              f (X )= 11           2X = 11 -3 = 8     X = 4 it is true also for all values of f(X).
 
            Now consider   g( X) = X2        X = ( 1, 2, 3, .......) result in g(X) = ( 1, 4, 9, ......)
 
              But for g(X) = ( 1, 4, 9 , .....)  result in  X (±1,±2, ±3) and so on.
             
                 Therefore by definition g(x) is not a one to one function.
 
      For a function to be one to one. It has to pass the vertical and horizontal test.
      From each point of the curve, if we draw a horizontal and vertical line it should have only one
      intersection point. 
 
      In above example of g(x) = 4 , drawing  Y = 4 intersects g(X) in 2 points of X=±2 . 
        
One to one
 
If you can draw a horizontal line that crosses the graph in more than one place, then the function is NOT one to one.  For example, a parabola is a u-shaped graph.  You can cross the graph in two places.  Any straight line is a one to one graph.