Answer:
g(x) is the square root of 4x^2 + 4x + 1.
Explanation:
We are given that f(x) = x^2 + 1 and f(g(x)) = 4x^2 + 4x + 2.
To find g(x), we need to substitute g(x) into the expression for f and simplify:
f(g(x)) = (g(x))^2 + 1 = 4x^2 + 4x + 2
Subtracting 1 from both sides, we get:
(g(x))^2 = 4x^2 + 4x + 1
Taking the square root of both sides (remembering to include both the positive and negative roots), we get:
g(x) = ±√(4x^2 + 4x + 1)
However, we need to choose the sign of g(x) such that f(g(x)) matches the given expression of f(g(x)) = 4x^2 + 4x + 2.
Let's try using the positive root first:
g(x) = √(4x^2 + 4x + 1)
Then we can find f(g(x)):
f(g(x)) = (g(x))^2 + 1 = 4x^2 + 4x + 2
This matches the given expression, so we can conclude that:
g(x) = √(4x^2 + 4x + 1)
Therefore, g(x) is the square root of 4x^2 + 4x + 1.