Final answer:
The student's question involves finding the composite functions (f ∘ g)(x) and (g ∘ f)(x), the product function (fg)(x), and determining the domains of these functions. Each operation has its own domain, which must consider the individual domains of f(x) and g(x).
Step-by-step explanation:
The question asks for the composition and product of two functions f(x) and g(x), and also requires to find the domains of these new functions resulting from the operations.
(f ∘ g)(x) is the composition of f and g, meaning we evaluate g and then apply f to the result. This is written as f(g(x)). The domain of this function consists of all x in the domain of g such that g(x) is in the domain of f.
(g ∘ f)(x) is the composition of g and f, meaning we evaluate f and then apply g to the result. This is written as g(f(x)). The domain of this function is all x in the domain of f such that f(x) is in the domain of g.
(fg)(x) is the product of f and g, calculated as f(x)*g(x). The domain of the product is the intersection of the domains of f and g, only including the x-values that are in both domains.
Finally, for the horizontal line function f(x) with the domain 0 ≤ x ≤ 20, the function is a constant value, and its domain is all real numbers between 0 and 20, inclusive.