Domain of a function

More precisely, given a function ${\displaystyle f:X\to Y}$, the domain of f is X. In modern mathematical language, the domain is part of the definition of a function rather than a property of it.

In the special case that X and Y are both sets of real numbers, the function f can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the x-axis of the graph, as the projection of the graph of the function onto the x-axis.

For a function ${\displaystyle f:X\to Y}$, the set Y is called the codomain: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of X is called its range or image. The image of ${\displaystyle f}$ is a subset of Y, shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. The restriction of ${\displaystyle f:X\to Y}$ to ${\displaystyle A}$, where ${\displaystyle A\subseteq X}$, is written as ${\displaystyle {f|}_A:A\to Y}$.

Natural domain

If a real function f is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, and the set of real numbers on which the formula can be evaluated to a real number is called the natural domain or domain of definition of f. In many contexts, a partial function is called simply a function, and its natural domain is called simply its domain.

Examples

• The function ${\displaystyle f}$ defined by ${\displaystyle f(x)=\frac{1}{x}}$ cannot be evaluated at 0. Therefore, the natural domain of ${\displaystyle f}$ is the set of real numbers excluding 0, which can be denoted by ${\displaystyle \mathbb{R}\setminus \{0\}}$ or ${\displaystyle \{x\in \mathbb{R}:x\ne 0\}}$.
• The piecewise function ${\displaystyle f}$ defined by ${\displaystyle f(x)=\{\begin{array}{ll}1/x& x\ne 0\\ 0& x=0\end{array},}$ has as its natural domain the set ${\displaystyle \mathbb{R}}$ of real numbers.
• The square root function ${\displaystyle f(x)=\sqrt{x}}$ has as its natural domain the set of non-negative real numbers, which can be denoted by ${\displaystyle {\mathbb{R}}_{\ge 0}}$, the interval ${\displaystyle [0,\mathrm{\infty })}$, or ${\displaystyle \{x\in \mathbb{R}:x\ge 0\}}$.
• The tangent function, denoted ${\displaystyle \tan }$, has as its natural domain the set of all real numbers which are not of the form ${\displaystyle \frac{\pi }{2}+k\pi }$ for some integer ${\displaystyle k}$, which can be written as ${\displaystyle \mathbb{R}\setminus \{\frac{\pi }{2}+k\pi :k\in \mathbb{Z}\}}$.