Find a formula for f^(-1)(x) if f(x)= ((x-1)/(x+1))^3

Question

Find f^-1(x)= 

Find (f^-1)'(x)= 

Answer 1

f(x)=((x-1)/(x+1))³

 

step 1: replace f(x) by y i.e.

            y = ((x-1)/(x+1))³

 

step 2: find the value of x in terms of y:for this make adjustments ,collect all the terms of x together on one side of equal to sign and then shift every thing except x to the other side.

             y= ((x-1)/(x+1))³

            y^1/3=  (x-1)/(x+1)

    y^1/3(x+1) = (x-1)

    xy^1/3 + y^1/3 =x-1

   xy^1/3 -x = -1 -y^1/3

   x(y^1/3-1) = -(1+y^1/3)

       x   =  - (1+y^1/3) /(y^1/3 -1)  

       x   = (1+y^1/3)/(1-y^1/3)   [taking - sign with term y^1/3-1 changes it  to  1-y^1/3]

step 3: replace x by f^-1(x) and y by x:

 

     f^-1(x) = (1+x^1/3)/(1-x^1/3)

 

 

For (f^-1)'(x) = d/dx (f^-1(x)) =d/dx[(1+x^1/3)/(1-x^1/3)]      

                                        =[ (1-x^1/3) d/dx(1+x^1/3) - (1+x^1/3) d/dx(1-x^1/3)]/(1-x^1/3)²

                                        =[ (1-x^1/3)((1/3)x^-2/3) -  (1+x^1/3)((-1/3)x^-2/3)]/(1-x^1/3)²