Inverses

Inverses

Example:  

The inverse of the relation

        (2,3), (4,5), (2,6), (4,6) 

is

        (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.

Example

Find the inverse of 

        y  =  2x + 1

Solution

1. We write
    
            x = 2y + 1
   
   
   

2. We solve:  
   
           x - 1  =  2y 
   
                     x - 1
           y  =                  
                       2
   
   

3.  We write 
   
                              x - 1
           f ^-1(x)   =                   
                                2
   

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.

Exercises

Find the inverse of

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

2. f(x)  =  3x³ - 2   
   
   

3. f(x)  =  3x - 4   
   
   

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.

Example

1. Determine if 

           y = 2x - 3

   is one to one
   

   Solution

   For any two values a and b if 

           y = 2a - 3 

   and 

           y = 2b - 3

   then

           2a - 3  =  2b - 3

           2a  =  2b

           a  =  b

   Hence this function is 1-1.

    

    

2. Determine if 

   y  =  x²

   is one to one
   

   Solution

   Now if  

           a²   =   b²  

   then 

           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.

 

Theorem 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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