my previous calculus teacher stressed finding the domain of a composite function; he stressed more of finding which areas were not part of the function by means of three circles and saying the function didn't pass to the other circle unless it met the range of the other function. The textbook mostly just says the domain of the composite function is the union of the domains of the two functions. Well, I'm just wandering why the domain of the composite functions is the union of the two functions domain and I think he said multiplied by the range of one of the functions(either inside or outside. Basically, the books are not to in depth with this, and i think I need a description of the whole process with maybe a proof of why the range comes into play.(I'm thinking maybe it has to do with the way the inverse functions switch range and domain?) thanks in advance.