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!
then when is
sqrt(x+1)=sqrtx + 1
at x=0, they are equal.
Why would x=0?? And how are they equal??
The question ask for the value(s) of X
To find all 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. Let's go step by step:
Step 1: Substitute f(x) = sqrt[x] and g(x) = x + 1 into the equation.
f(g(x)) = g(f(x))
sqrt[x + 1] = f(x) + 1
Step 2: Substitute f(x) = sqrt[x] into the equation.
sqrt[x + 1] = sqrt[x] + 1
Step 3: Square both sides of the equation to eliminate the square root.
(x + 1) = (sqrt[x] + 1)^2
Step 4: Expand and simplify the equation.
(x + 1) = x + 2sqrt[x] + 1
Step 5: Cancel out the 1's on both sides of the equation.
x = 2sqrt[x]
Step 6: Square both sides of the equation again to eliminate the square root.
x^2 = (2sqrt[x])^2
x^2 = 4x
Step 7: Move all terms to one side of the equation and set it equal to zero.
x^2 - 4x = 0
Step 8: Factor out x from the equation.
x(x - 4) = 0
Step 9: Apply the Zero Product Property, which states that if a product of factors equals zero, then at least one of the factors must be zero.
x = 0 or x - 4 = 0
Step 10: Solve for x in both equations.
For x = 0, we get g(x) = 0 + 1 = 1. Substituting into f(x) gives us f(1) = sqrt[1] = 1. Therefore, x = 0 is a valid solution.
For x - 4 = 0, we get x = 4. Substituting into g(x) gives us g(4) = 4 + 1 = 5. Substituting into f(x) gives us f(4) = sqrt[4] = 2. Therefore, x = 4 is also a valid solution.
Step 11: So, the possible values of x for which f(g(x)) = g(f(x)) are x = 0 and x = 4.
I hope this explanation helps! Let me know if you have any further questions.