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 find all the possible values of x for which f(g(x)) = g(f(x)), we need to substitute the expressions for f(x) and g(x) into the equation and solve for x.
First, let's find f(g(x)):
f(g(x)) = f(x + 1) = sqrt(x + 1)
Now, let's find g(f(x)):
g(f(x)) = g(sqrt(x)) = sqrt(x) + 1
Setting these two expressions equal to each other, we get:
sqrt(x + 1) = sqrt(x) + 1
To solve this equation, we need to isolate the radical term on one side and the constant term on the other side. Here's how we can do that:
1. First, let's square both sides of the equation to eliminate the square root:
(sqrt(x + 1))^2 = (sqrt(x) + 1)^2
x + 1 = x + 2sqrt(x) + 1
2. Next, let's simplify the equation:
x + 1 = x + 2sqrt(x) + 1
The x terms cancel out, and we are left with:
2sqrt(x) = 0
3. Now, we isolate the radical term by subtracting x from both sides:
2sqrt(x) - x = 0
4. To continue simplifying, let's factor out sqrt(x):
sqrt(x)(2 - sqrt(x)) = 0
According to the Zero Product Property, either sqrt(x) = 0 or (2 - sqrt(x)) = 0. Let's solve these two equations separately:
For sqrt(x) = 0:
Square both sides to eliminate the square root:
(sqrt(x))^2 = 0^2
x = 0
For 2 - sqrt(x) = 0:
Add sqrt(x) to both sides:
2 = sqrt(x)
Square both sides:
(2)^2 = (sqrt(x))^2
4 = x
So, we have two possible values for x: x = 0 and x = 4.
To summarize, the possible values of x for which f(g(x)) = g(f(x)) are x = 0 and x = 4.