Final answer:
The function f(g(x)) = 22 + 2√(2 - x) has a domain of all real numbers x such that x ≤ 2. For g(f(x)), since it involves the square root of a negative number, there is no real domain.
Step-by-step explanation:
Composition of Functions and Domain
To solve Part B, we will find f(g(x)) which means we'll substitute g(x) into f(x). Given that f(x) = 22 + 2 and g(x) = √(2 - x), we substitute the g(x) function into the f(x) to get f(g(x)) = 22 + 2√(2 - x). The domain of the composed function f(g(x)) is determined by the range of values x can take such that the argument of the square root, (2 - x), is non-negative. Therefore, the domain of f(g(x)) is all real x such that x ≤ 2.
For Part C, we find g(f(x)), substituting f(x) into g(x). Thus, g(f(x)) = √(2 - (22 + 2)) = √(-22). This expression involves taking the square root of a negative number, which is not possible within the real number system, hence there is no real domain for g(f(x)).