Definition of Real-Valued Functions, Domain, Codomain

Domain and Codomain

When discussing functions, the domain is the set of elements for which a function is defined, and the codomain, also sometimes referred to as the range, is the set of elements that comprise a function's output.

For example, we can say that the domain of the function $y = 2x$ contains all real numbers - that is, we can plug in any real number and obtain a valid output from the function. The domain of a function can be expressed in several ways. We can write the domain of $y = x$ informally as "all real numbers", or we can also express it as $(-\infty, +\infty)$, the set of all numbers that lie between $-\infty$ and $+\infty$.

Similarly, we can say that the codomain of the function $y = 2x$ also contains all real numbers. For every number we plug into the function, we get a real number as an output. We can also express the codomain of $y = 2x$ as "all real numbers" or $(-\infty, +\infty)$.

It is generally assumed that the domain and codomain exclude nonreal numbers, especially when working with the $x,y$ (two-dimensional) Cartesian plane.

Real-Valued Functions

With these definitions for domain and codomain, we can then say a function is real-valued or simply real if its codomain is comprised entirely of real numbers.