Normally in algebra we deal with continuous functions that are described by an infinite domain and a "recipe." However, it is illustrative to define functions with a finite domain and range and are defined explicitly.

Consider the two functions:

f(x) 0   →   0 1   →   1 2   →   4 3   →   9

g(x) 1   →   2 2   →   4 3   →   6 4   →   8

The domain of f is the set {0,1,2,3} with a range {0,1,4,9}. The domain of g is the set {1,2,3,4} with a range {2,4,6,8}.

The composition, g(f(x)) is:

g(f(x)) 1   →   1   →   2 2   →   4   →   8

or

g(f(x)) 1   →   2 2   →   8

We cannot "start" with 0 or 3 because f maps them to a 0 and 9, respectively, which are not in the domain of g.

Similarly, the 2 and 3 that are in the domain of g cannot be part of the composition because they are not in the range of f.

Therefore, the domain of the composition, g(f(x)) is the set {1,2} and its range is {2,8}.

The domain of the composition, g(f(x)), must be in the domain of f and it contains only those values that map to the domain of g.

The range of the composition, g(f(x)), must be in the range of g and it contains only those values that are mapped from values in the domain of g.