Final answer:
Assuming typo correction, fog is calculated by substituting g into f, yielding 4(x + 1)² + 1. For gof, substitute f into g, giving (4x² + 1) + 1. The domains for all functions in interval notation are (-∞, ∞).
Step-by-step explanation:
To find the composite functions fog (f ○ g) and gof (g ○ f), we first need to define the functions f(x) and g(x). However, it seems there was a typo in the question. Assuming that f(x) = 4x² + 1 and g(x) = x + 1, we can proceed as follows:
For fog, we substitute g(x) into f(x):
f(g(x)) = f(x + 1) = 4(x + 1)² + 1.
For gof, we substitute f(x) into g(x):
g(f(x)) = g(4x² + 1) = (4x² + 1) + 1.
The domain of f is all real numbers since it's a polynomial function. Similarly, the domain of g is all real numbers because it's a linear function. For the composite functions, domain of fog and domain of gof, their domains are also all real numbers because composite functions maintain the domain restrictions of their inside functions, provided that the outside function does not introduce additional restrictions, which is not the case here.
Therefore, in interval notation, the domain for all the functions is (-∞, ∞).