Final answer:
The domain of the function f(x) / g(x) is all real numbers except zero and three, as those are the values that would result in the denominator being zero for each respective function.
Step-by-step explanation:
To determine the domain of the composite function f(x) / g(x), we must consider the domains of both f(x) and g(x) separately and then find the intersection. For the function f(x) = x² − 1 / x, we must ensure that the denominator is not equal to zero, which means that x cannot be zero. Therefore, the domain of f(x) is all real numbers except zero.
For the function g(x) = x + 2 / x − 3, the denominator cannot be equal to zero, hence x cannot be 3. So, the domain of g(x) is all real numbers except three.
To find the domain of the function f(x) / g(x), we need the domains where both functions are defined, thus the domain of f(x) / g(x) is all real numbers except zero and three.