The inverse of a relation R is the relation consisting of all ordered pairs (y,x) such that (x,y) belongs to R



The inverse of the relation

        (2,3), (4,5), (2,6), (4,6) 


        (3,2),  (5,4), (6,2), (6,4)

Generally we switch the roles of x and y to find the inverse.

For functions, we follow the steps below to find the inverse:


  • Step 1:  Switch the x and y.


  • Step 2:   Solve for y.


  • Step 3:  Write in inverse notation.



Find the inverse of 

        y  =  2x + 1




  1. We write
             x = 2y + 1

  2. We solve:  

            x - 1  =  2y 

                      x - 1
            y  =                  

  3.  We write 

                               x - 1
            f -1(x)   =                   

Notice that the original function took x, multiplied by 2 and added 1, while the inverse function took x, subtracted 1 and divided by 2.  The inverse function does the reverse of the original function in reverse order.



Find the inverse of


  1.               x - 1
    f(x)  =                
                  x + 1

    (x + 1) / (x - 1)

  2. f(x)  =  3x3 - 2   

    cubeRt(x/3 + 2/3)

  3. f(x)  =  3x - 4   

    (x +4)/3



Graphing Inverses

To graph an inverse we imaging folding the paper across the y = x line and copy where the ink smeared in the other side.  





One to One Functions

A function  y = f(x) is called one to one if for every y value there is only one x value with y = f(x).  That is, each y value comes from a unique x value.


  1. Determine if 

            y = 2x - 3

    is one to one


    For any two values a and b if 

            y = 2a - 3 


            y = 2b - 3


            2a - 3  =  2b - 3

            2a  =  2b

            a  =  b

    Hence this function is 1-1.



  2. Determine if 

    y  =  x2

    is one to one


    Now if  

            a2   =   b2  


            a   =   b

    In particular 

            a = -b 

    is a viable solution, for example 

            12  = (-1)2

    hence this function is not 1-1. 

The Horizontal Line Test

If the graph of a function is such that every horizontal line passes through the graph at at most one point then the function is 1-1.  

The graph below is the graph of a 1 -1 function since every horizontal line crosses the graph at most once.


However, the graph below is not the graph of a 1 -1 function, since there is a horizontal line that crosses the graph more than once.




If a function is 1-1, then its inverse is also a function

An application of this is if we want a computer to find an inverse function, then we first have the computer check to see if the function is one to one, then have it proceed to find the inverse.


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