Final answer:
To find (f ∘ g)(x), substitute g(x) into f(x). Then, for (f ∘ g)(-3), replace x with -3 in the composed function and simplify to find the value.
Step-by-step explanation:
The student has asked to perform the indicated compositions f(g(x)) where f(x) = 3x² + 2x, and g(x) = -3x + 1. To find (f ∘ g)(x), you substitute g(x) into f(x). That is, replace every instance of x in f(x) with g(x).
The composition f(g(x)) is therefore f(-3x + 1), which equals 3(-3x + 1)² + 2(-3x + 1). Simplify this expression to find the composed function (f ∘ g)(x).
Next, we evaluate (f ∘ g)(-3). To do this, replace x with -3 in the expression we just found for (f ∘ g)(x). After substituting and simplifying, we will get the numerical value for (f ∘ g)(-3).
To find (fog)(x), we need to replace 'x' in the function f(x) with the function g(x).
First, substitute g(x) into f(x):
f(g(x)) = 3(g(x))^2 + 2(g(x))
= 3(-3x + 1)^2 + 2(-3x + 1)
= 3(9x^2 - 6x + 1) - 6x + 2
= 27x^2 - 18x + 3 - 6x + 2
= 27x^2 - 24x + 5
Now, to find (fog)(-3), we substitute -3 into the function (fog)(x):
(fog)(-3) = 27(-3)^2 - 24(-3) + 5
= 27(9) + 72 + 5
= 243 + 72 + 5
= 320