Domain of a Function

Definition

In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation f: X → Y, and is alternatively denoted as dom(f). Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.

A domain is part of a function f if f is defined as a triple (X, Y, G), where X is called the domain of f, Y its codomain, and G its graph.

A domain is not part of a function f if f is defined as just a graph. For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form f: X → Y.

For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).

If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the x-axis. The domain of a function f can be thought of as the set of all x values that can be plugged into f(x) that return a valid output. For example, if we have a function g(x) in the Cartesian plane, the domain is all of the x values such that g(x) is a real number.

Examples:

• Let $S$ be a set of ordered pairs such that $S=\{(1,2),(2,3),(4,7),(13,9),(-20,0)\}$. The domain is the set of all x values of $S$, so the domain is $\{-20,1,2,4,13\}$.
• The domain of $g(x)=1/x$ is all real numbers EXCEPT 0, since 1/0 is not defined.
• The domain of $h(x)={\sqrt {x}}$ is $[0,\inf )$, since ${\sqrt {x}}$ is only defined when $x$ is nonnegative (that is, when $x$ is greater than or equal to 0).

Resources and Examples

• Domain of a Function, Wikipedia
• Domain and Range: Basic Idea, patrickJMT
• Finding Domain with an Equation, Lumen Learning
• Finding Domain and Range with Graphs, Lumen Learning
• Finding the Domain Algebraically, patrickJMT
• Finding Domain and Range of a Piecewise Function, patrickJMT
• How to Find Domain + Example Problems, Math Culus