Suppose that f(x)=x^3 and g(x)=1/x+1. For what value(s) of x is (f∘g)(x)=(g∘f)(x)

Answer 1

f(x) = x^3

g(x) = 1/(x+1)

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

gof(x) = 1/(x^3+1)

fog(x) = gof(x)

1/(x^3+3x^2 +3x + 1) = 1/(x^3 + 1)

x^3 + 3x^2 +3x + 1 = x^3 + 1

3x^2 +3x = 0

x(x+1) = 0

x = 0 (x=-1 would be another solution, except it makes g(x) undefined, 1/(-1+1) = 1/0)