Final answer:
The student needs to find the composite functions f ◦ g and g ◦ f given f(x) = 2x^2 and g(x) = x + 3. To do this, substitute g(x) into f for f ◦ g, and substitute f(x) into g for g ◦ f, then simplify.
Step-by-step explanation:
The student is asking to find the composite functions f ◦ g and g ◦ f, where f(x) = 2x2 and g(x) = x + 3. Composite functions involve applying one function to the result of another function.
To find f ◦ g, we first apply g to x and then apply f to the result of g(x):
f(g(x)) = f(x + 3) = 2(x + 3)2
To find g ◦ f, we first apply f to x and then apply g to the result of f(x):
g(f(x)) = g(2x2) = 2x2 + 3
In each case, we substituted the inside function into the outside function and simplified where possible.