One to One functions

 

A one-to-one correspondence exists when all of the elements of  set A

 map to exactly one element of set B and vice versa.

 

          

                  This is a one-to-one correspondence.

 

 

 

          

         This is a function, but not a one-to-one correspondence.

 

 

 

 When a one-to-one correspondence exists,

 the function that maps from set B to set A is called the inverse of f.

 The inverse of a function f(x) is denoted f^-1(x)

 

 

 

           

 

        For each element,   f(a) = b     and        f^-1 (b) = a

 

           The domain of f is the range of f^-1

            The range of f is the domain of f^-1

 

 

                   f^-1 (f(x)) = f (f^-1 (x)) = x                 

 

 

 

To inverse a function ( which must be in a one  one correspondence)

reflect it in the line y = x.

 

Example

 

 Find the inverse of the function   f(x) = 2x + 2   

 

    f(x) = 2x + 2    has graph     y = 2x + 2

    f(0) = 0 + 2 = 2    so the point (0,2) is on the line y = 2x + 2.

    f(2) = 4 + 2 = 6    so the point (2,6) is on the line y = 2x + 2.

 

 

    If this function is reflected in the line y = x,

    the values for x and y are swapped, since y = x and x = y.

 

   So the point (0,2) becomes ( 2,0)

   and the point ( 2,6) becomes ( 6,2)

 

The equation of the reflected line can now be found:-

 

      

 

 

 

 

·       Algebraic method

 

Example

Find the inverse of the function f(x) = 2x + 2

 

f(x) = 2x + 2    has graph     y = 2x + 2

Make x  the subject of the formula   y = 2x + 2

 

                y = 2x + 2

                y - 2 = 2x

                (y - 2) = x

so            x =  (y - 2)

 

Swap the letters (because of the reflection y = x)

 

             y =  (x - 2)

             y = x  1

 

 so  f ^-1 (x) = x  1

 

 

Example

 

 

 

Example