Final answer:
The domain of g(f(x)) is all real numbers except x=-3, x=0, and x=1.
Step-by-step explanation:
The domain of the composite function g(f(x)) can be found by considering the restrictions of both f(x) and g(x).
Since f(x) is defined as (x+3)/(x-1), the only restriction is that x cannot equal 1. Therefore, the domain of f(x) is all real numbers except x=1.
Next, we consider g(x), which is defined as (x²-2)/x. The only restriction here is that x cannot be equal to 0, since division by zero is undefined. Therefore, the domain of g(x) is all real numbers except x=0.
When we compose the functions g(f(x)), we need to ensure that the input to g(x) is within its domain. Since f(x) has a domain of all real numbers except x=1, the only restriction on the domain of g(f(x)) is that f(x) cannot equal 0. Therefore, the domain of g(f(x)) is all real numbers except if f(x)=0, which means x+3=0, leading to x=-3.