The domain and range of a function designate the values at which the function exists. In a composite function, one function is being applied to another. You must determine values at which the first function does not exist and values for which the first function creates values where the second function does not exist. This process usually takes only a few minutes to complete.
Determine the order the functions are applied
Examine the separate functions, often labeled f(x) and g(x). Figure out whether function f is being applied to the result of g(x) or the other way around. Composite functions are usually designated by a small 'o' symbol, such as º in the case of (g º f)(x). This means that you take function g of the result of the function f(x). This can be written as g(f(x)).
Figure out the domain of the first function
The first or inside function happens first, so any values at which the first function does not exist do not exist for the composite function. For example, if the first function is f(x)=-x/(x-5), then x could not equal 5, because that would create a 0-denominator. The range of f(x) is all real numbers except 5.
Figure out the domain of the composite function
If f(x)=-x/(x-5) and g(x)=?x, then g(f(x))=?f(x)=?(-x/(x-5)). The domain of g(x) is all real non-negative numbers. However, the domain of (g(f(x)) is all real numbers except for those at which f(x) is negative. f(x) is negative whenever x is greater than 5 or less than 0. The domain of (g(f(x)) is all real numbers greater than or equal to 0 and less than 5. Any restrictions in the domain of the first function must still be applied to the composite function.