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

Question

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

Answer 1

ƒ(x) = y = ((x-1)/(x+1))³

 

To find f^-1 replace x's with y's and y's with x's and isolate y

 

x = ((y-1)/(y+1))³

 

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

 

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

 

yx^1/3+x^1/3 = y-1

 

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

 

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

 

Derivative of f^-1

 

f^-1'(x) = ((1-x^1/3)(x-^2/3/3)-(1+x^1/3)(-x^-2/3/3)/(1-x^1/3)²

 

Simplification of the above expression yields 

 

f^-1(x) = 1/(3x^2/3(1-x^1/3)²)