Inverse Function

If a set  obtained  by  interchanging  the  co-ordinates   of 

each ordered pair  of  a function  is  a  function  , then  the given  function  is  an  invertible  function  and  the   new  function  is  the  inverse  function  .

A  function  has  inverse  if and only if it  is  a  one-to-one  

function .

 E.g.

*

{(1,2),(3,4),(5,1)} → {(2,1),(4,3),(1,5)}

It is  a  invertible  function and its inverse  function  

{(2,1),(4,3),(1,5)}.

*

{(1,2),(3,2),(5,1)} →  {(2,1),(2,3),(1,5)}

It is a function but not invertible  function  . So  Its can not  do  inverse .

Definition:  Lets f  be  a  One-to-One  function with  domain X and  range  Y .The  inverse  of  f  is  the  function  g  with  domain  Y and  Range X  for which

 

        f(g(x)) = x  for every x in Y   

  and  g(f(x)) = x for every x in X

So f  is not  One-to-One ,then it has no inverse function.

Procedure of finding  f^-1(x)

    1) Check whether  the  function  is  One-to-One

    2) Write the function y=f(x)

    3) Solve for x (in terms of y)

    4) Interchange x and y

    5) Write y= f^-1(x)

Properties :

i)                 Domain of f = Range of f^-1

ii)              Range of f = Domain of f^-1

iii)           f^-1  is One-to-One.

iv)           Inverse of f^-1 is  f

Example :

y = f(x) =(x+1) / (2x-5)

Domain of f = R – {5/2 }

Here, y = (x+1)/(2x-5)

      => 2xy-5y=x+1

      => 2xy-x=5y+1

      => x(2y-1)=5y+1

      So x = (5y+1) / (2y-1)

Interchanging x and y , we get , 
       y = (5x+1) / (2x-1) = f^-1(x)

Domain of f = R – {1/2 }

Range of f = R – {5/2}

Graphs of f and f^-1

The   graph  of  f and f^-1 are reflections  of  each other  in the  line  y = x . Tn the  other  words , the graphs  of  f and  f^-1  are  symmetric  with  respect  to  the  line  y=x .

Such  as ,  y = x³