Thread: Domains of functions

Given f(x) = x^3 and g(x) = 1/(1+x^2).

What is g(x)/f(x) and what is the domain?

I've figured out that g(x)/f(x) = 1/(x^3 + x^5) but I'm unsure of what the domain is. My lecturer stated that the domain of a composite function will be the same as the domain of its original functions. Both f(x) and g(x) have domains of (-infinity, infinity) so that would mean g(x)/f(x) also has a domain of (-infinity, infinity). But this does not make sense to me because g(x)/f(x) is undefined when x = 0 as we cant divide by 0.

Could someone please clear this up for me? I'll be very thankful.

g(x)/f(x) = 1/x^3(x^2+1). The only undefined way is at x=0. Thus, the domain is (this reads as all points minus 0).