Final answer:
The domain of the composite function (f∘g)(x) with f(x)=1/(x−2) and g(x)=√(x+4) is all real numbers x where x≥−4, excluding x=−2; hence, the domain is [−4, −2) ∪ (−2, ∞).
Step-by-step explanation:
To find the domain of the composite function (f∘g)(x), we first need to determine the domain of g(x) and then apply it to f(x). The function g(x) = √(x+4) is defined for all values of x where x+4 ≥ 0, because the square root of a negative number is not a real number. Therefore, the domain of g(x) is [−4, ∞). The function f(x) = 1/(x−2) is defined for all values of x except for x = 2 because division by zero is undefined.
When we compose f and g to get (f∘g)(x) = f(g(x)), we must ensure that g(x) does not produce a result that would be undefined in f(x). Since g(x) could be any number in [−4, ∞), we check where f(g(x)) becomes undefined, which would be when g(x) = 2. This happens when x+4 = 2, i.e., x = −2. So the only restriction on our domain from f is that x ≠ −2.
Combining these, the domain of (f∘g)(x) is all real numbers x such that x ≥ −4 but x ≠ −2. Hence, the domain in interval notation is [−4, −2) ∨ (−2, ∞).