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

Question

f(x) = x³ + 1

 

 

g(x) = ^3√x − 1

Answer 1

f(x) = x³+1   g(x) = ³√(x - 1)

 

f(x) and g(x) are inverses if f(g(x)) = g(f(x)) = x

 

f(g(x)) = f(³√(x-1)) =( ³√(x-1))³ + 1 = x-1+1 = x

 

g(f(x)) = g(x³+1) = ³√[(x³+1)-1] = ³√(x³) = x

 

Since f(g(x)) = g(f(x)) = x, f and g are inverses of each other.  

 

Answer 2

For f(x), switch places with x and y. Then solve for y.

 

y = g(x)  means f(x) and g(x) inverses to each other.