Use the Inverse Function Theorem to show that f and g are in

Use the Inverse Function Theorem to show that f and g are inverses

of each other.

f(x) = x^3 + 1 g(x) = 3square root x - 1

Solution

By Inverse Function Theorem : (F^-1)\'F(x)) = 1/F\'(x)

F(x) = x^3 +1

differentiate F(x) : F\'(x) = 3x^2

differentiate F^-1(x) = g(x): g\'(x) = (x-1)^(-2/3)/3

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

(F^-1)\'F(x)) = 1/F\'(x)

LHS = 1/3x^2

RHS = 1/3x^2

Hencve proved by inverse function f(x) and g(x) are inverse pf each other