Final answer:
The domain of the composite function f(g(x)) is determined by first finding the domains of g(x) and f(x) individually. The domain of g(x) is [–4, ∞) and the domain of f(x) is (–∞, 4) ∪ (4, ∞). Accounting for these restrictions, the domain of f(g(x)) is [–4, ∞), excluding the point where g(x) would make the function undefined, and the correct option is Option 3 [–4, ∞).
Step-by-step explanation:
To find the domain of the composite function f(g(x)), we first need to determine the domain of each individual function and then take into account how one function affects the other. The function g(x) is equal to the square root of the quantity x + 4. Since we cannot take square roots of negative numbers (assuming we are dealing with real numbers), the domain of g(x) is [–4, ∞). Next, we look at the function f(x) which is 1/(x - 4). We cannot divide by zero, so x cannot be 4. This restriction creates a domain for f(x) which is (–∞, 4) ∪ (4, ∞). For the composite function f(g(x)), x values must be within the domain of g(x), which is then plugged into f(x). Since g(x) cannot be 4 because it would cause division by zero in f(x), the value of g(x) that would make f(g(x)) undefined is when g(x) = 4, which happens when x = 0. Therefore, 0 should be excluded from the domain of g(x) when considering the domain of f(g(x)). This results in the domain being [–4, 0) ∪ (0, ∞), which corresponds to Option 3 [–4, ∞).