G(x)=4x^3+5 find the inverse function and also verify the function

Question

G(x)=4x^3+5 find the inverse function and also verify the function

Answer 1

Let y = 4x³+5

 

Switch x and y:  x = 4y³+5

 

Solve for y:  4y³ = x-5

 

                   y³ = (x-5)/4    So, y = [(x-5)/4]^1/3

 

                                            G^-1(x) = [(x-5)/4]^1/3

 

Verification:  Show that G(G^-1(x)) = G^-1(G(x))=x

 

           G(G^-1(x)) = G([(x-5)/4]^1/3) = 4([(x-5)/4]^1/3)³ + 5

 

                                                 = 4([(x-5)/4]) + 5 = x-5+5 = x

 

           G^-1(G(x)) = G^-1(4x³+5) = [(4x³+5-5)/4]^1/3

 

                                            = [4x³/4]^1/3 = [x³]^1/3 = x