Final answer:
To find g(x), we look at the composition g(f(x)) and substitute x with the inverse operation of f(x), simplifying to get the quadratic form. After substitution and simplification, we can determine the coefficients a, b, and c for g(x).
Step-by-step explanation:
The student has provided the composition of two functions, which means g(f(x)) has already been evaluated at f(x) = 2x + 1, resulting in 4x² + 4x + 3. To find g(x), we must express this composition in terms of an input that has not been transformed by f(x), effectively reversing the transformation of f(x).
To do this, we need to take the function g(f(x)) = 4x² + 4x + 3 and replace 'x' with the inverse operation of f(x), which would be (x - 1)/2. However, since we want g(x) in the form ax² + bx + c, it's easier to substitute x directly into f(x) and then into g(f(x)), reasoning backwards to get the form of g(x). Since f(x) = 2x + 1, we can substitute x for (x - 1)/2 in g(f(x)) to find g(x).
Thus, g(x) = g((x - 1)/2) = 4((x - 1)/2)² + 4((x - 1)/2) + 3. The next step is to expand and simplify this equation to get it in the standard quadratic form. Upon simplifying, we would find that the coefficients a, b, and c for g(x) in ax² + bx + c directly correspond to the expanded version of our equation after substituting and simplifying. The specific values for a, b, and c will be determined after the expansion and collection of like terms.