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!

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)

Where did you get the x+2 from?

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

To determine all possible values of x for which f(g(x)) = g(f(x)), we need to evaluate the composition of the two functions f and g and equate them.

Step 1: Find f(g(x)):
To find f(g(x)), we substitute g(x) into the function f(x):
f(g(x)) = sqrt[g(x)] = sqrt[x + 1]

Step 2: Find g(f(x)):
To find g(f(x)), we substitute f(x) into the function g(x):
g(f(x)) = f(x) + 1 = sqrt[x] + 1

Step 3: Equate f(g(x)) and g(f(x)):
We set the expressions for f(g(x)) and g(f(x)) equal to each other:
sqrt[x + 1] = sqrt[x] + 1

Step 4: Simplify the equation:
To eliminate the square roots, we will square both sides of the equation:
(x + 1) = (sqrt[x] + 1)^2

Expanding the right side:
(x + 1) = x + 2*sqrt[x] + 1

Simplifying the equation further:
x + 1 = x + 2*sqrt[x] + 1

Subtracting x from both sides:
1 = 2*sqrt[x]

Dividing both sides by 2:
1/2 = sqrt[x]

Step 5: Square both sides again to remove the square root:
(1/2)^2 = (sqrt[x])^2
1/4 = x

So, the possible values of x that satisfy f(g(x)) = g(f(x)) are x = 1/4.