Given a function , the set B is called the codomain of f.
The codomain is not to be confused with the range f(A), which is in general only a subset of B.
Let the function f be a function on the real numbers:
The codomain of f is R, but clearly f(x) never takes negative values, and thus the range is in fact the set R+—non-negative reals, i.e. the interval [0,∞):
One could have defined the function g thus:
While f and g have the same effect on a given number, they are not, in the modern view, the same function since they have different codomains.
The codomain can affect whether or not the function is a surjection; in our example, g is a surjection while f is not.