How do I determine the domain and range of a composite function, fg(x) ?

Firstly I would state the rule that for any composite function fg, its domain is always the domain of g. If the student would like to know why this is the case, I would simplify the expression fg(x) to f(g(x)) and show by counter-examples that the funtion f(g(x)) can only ever exist if the function g(x) is able to produce a range of numbers from the domain. (range of g is a subset or an equal set to the domain of f).
Next I would explain the range of fg is dependant of the funtions themselves. An expression would need to be derived for fg, and hence using the known domain of g, the range of fg could be determined. This would best be explained by example e.g. f(x) = (x+4)^0.5 , 2(x)^2 - 3