Given the functions f(x)=sqrt[x] and g(x)=x+1, determine all possible values of x for which f(g(x)) = g(f(x)). Show steps please, thanks a lot!

Answer 1

f(g(x)) = f(x+1) = √(x+1)

g(f(x)) = g(√x) = √x + 1

so √(x+1) = √x + 1
by observation, I can see that x=0 is a solution, ....

square both sides
x+1 = x + 2√x + 1
0 = 2√x
x = 0

(my intuition was correct)

Answer 2

Where did you get the x+2 from?

Answer 3

(sqrt x + 1)^2 = x + 2 sqrt x + 1