Domain Question

[sjaguar13]

If f(x)=x/(x-1) and g(x)=1/(x+3) what is the domain and range of (f.g)(x) and (f/g)(x)? Are the domains the same?

I got:
Domain of f is all x not equal to 1 or -3. The range is all real numbers.
x/(x-1)(x+3)

Domain of g is all x not equal to 1, but it can't be equal to -3 either because of f (this is the part I am not really sure about). The range is all real numbers.
x(x+3)/(x-1)

Yes, the domains are the same (Right?)

[Fredrik]

The domain of f is "x not equal to 1". The domain of g is "x not equal to -3 ". Because of that, neither 1 nor -3 can be a part of the domain of fg or f/g. It doesn't matter that f(x)/g(x)=x(x+3)/(x-1) for all x in the domain of f/g. You don't even need to know that. You only need to know that 0 is not in the range of g.

This is easy to understand if you just remember that the definition of the product function h=f/g is "h(x)=f(x)/g(x) for all x that are members of both domains, except x such that g(x)=0".

If you know this definition, the problem is trivial. If g is a function that doesn't have 0 in its range, the two domains must be the same.

The range of both functions is the set of real numbers.
.