Final answer:
To find the function g(x), we divide the composite function (fg)(x) by the given function f(x). Doing so, we determine that g(x) equals 2x minus 1, making b) g(x) = 2x - 1 the correct answer.
Step-by-step explanation:
This is asking about finding a function g(x) when the composite function (fg)(x) and the function f(x) are known. To find g(x), we need to express (fg)(x) in terms of f(x). The given f(x) = 2x + 3 and (fg)(x) = 4x² + 1. The composite function (fg)(x) is the result of multiplying the function f(x) by g(x). Knowing this, we proceed by dividing (fg)(x) by f(x) to isolate g(x): (fg)(x) / f(x) = g(x), ((4x² + 1) / (2x + 3)) = g(x), (2x * 2x + 1) / (2x + 3) = g(x). When we simplify the equation, we get g(x) = 2x - 1. Therefore, the correct answer is option b) g(x) = 2x - 1.