characteristics of functions

Question

given the functions f(x)x+2 and g(x)= x^{2 determine all values of x for which f(g(x))=g(f(x))}

Answer 1

f(x) = x+2      g(x) = x²

 

f(g(x)) = f(x²) = x²+2      g(f(x)) = g(x+2) = (x+2)²

 

If f(g(x)) = g(f(x)), then x²+2 = (x+2)²

 

So, x² + 2 = x² + 4x + 4

 

            2 = 4x + 4

 

           -2 = 4x        x = -1/2