The composition of two functions is a single function that, for a given initial value, takes the result of one function and gives it as imput to the second. The tricky part is to determine the appropriate domains and ranges.

We start with with a function, f(x) with a domain D_f and range R_f. The second function g(x) also has a domain D_gand range R_g. In general only some of the values in the domain of f will produce values that lie in the range of g. This smaller set will become the domain of the composite function g(f(x)). All the possible resultant values that lie in the range of g will become the range of the composite function.

Note that the domain of the composite function cannot be larger than the original domain of f.