Domain (function)

The domain of a function is the set of values on which that function is defined.

Typically, a function given by a particular rule might have many possible domains. For instance, the function $f(x) = x^2$ can take a wide variety of domains, if we were to assign it one.

(If we choose as its codomain the nonnegative real numbers, for instance, the domain could be the integers, the rational numbers, all of the real numbers, or many other sets. However, in this case the domain could not be the complex numbers, since some complex numbers have squares which are not nonnegative real numbers and so are not in our codomain.)

As an alternative example, if we take the function $f(x) = \frac1x$, mapping to the real numbers, our domain could be the set of all reals except zero, $\mathbb{R}-\{0\}$, but could not be all of the real numbers because $\frac10$ is not defined.

Function agreement

Given two functions $f$ and $g$ with different, but overlapping, domains, $A$ and $B$, respectively, we say that the functions agree on their shared domain, if for every $x$ in the domains of both $f$ and $g,$ $f(x) = g(x).$