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!
To determine the possible values of x for which f(g(x)) = g(f(x)), we need to find the values that satisfy this equation.
First, let's calculate f(g(x)):
f(g(x)) = f(x + 1)
The function f(x) is defined as the square root of x, so substituting x + 1 into f(x), we get:
f(g(x)) = sqrt[x + 1]
Now, let's calculate g(f(x)):
g(f(x)) = g(sqrt[x])
The function g(x) is defined as x + 1, so substituting sqrt[x] into g(x), we get:
g(f(x)) = sqrt[x] + 1
Now we can equate the two expressions:
sqrt[x + 1] = sqrt[x] + 1
To solve this equation, we need to square both sides:
(x + 1) = (sqrt[x] + 1)^2
x + 1 = x + 2sqrt[x] + 1
Simplifying the equation further:
x + 1 - x - 1 = 2sqrt[x]
2sqrt[x] = 0
To find the possible values of x, we can square both sides again to eliminate the square root:
(2sqrt[x])^2 = 0^2
4x = 0
x = 0
So the only possible value of x is 0. Therefore, the equation f(g(x)) = g(f(x)) holds true only when x = 0.