To find the domain and range of composite functions, first determine the domain of the inner function g(x) and then its range, which will inform the domain of the outer function f(x). The composite function f(g(x)) is only valid if the range of g(x) is a subset of the domain of f(x). The range of the composite function is derived from the range of f, restricted to the values that g can output. When calculating the range of combined functions, such as f(x) + g(x), consider how each function's range contributes to the overall output, especially at critical points like domain boundaries. Understanding these principles is essential for accurately determining the domains and ranges of composite functions.