Final answer:
To find the composite function (f ∘ g)(x), substitute g(x) into f(x) to get (f ∘ g)(x) = f(2x - 3). Expanding and simplifying yields the answer: (f ∘ g)(x) = 4x² - 6x + 1.
Step-by-step explanation:
The question asks for the composite function of f(g(x)), often denoted as (f ∘ g)(x), using the given functions f(x) = x² + 3x + 1 and g(x) = 2x - 3. To find this, we substitute the entire function g(x) for every instance of x in the function f(x). The calculation is as follows:
(f ∘ g)(x) = f(g(x)) = f(2x - 3)
After substituting, you apply the operations of the function f:
(f ∘ g)(x) = (2x - 3)² + 3(2x - 3) + 1
Expanding the squared term and simplifying, we obtain:
(f ∘ g)(x) = 4x² - 12x + 9 + 6x - 9 + 1 = 4x² - 6x + 1
The simplified form of the composite function (f ∘ g)(x) is therefore 4x² - 6x + 1.