# Domain of a function

## Mathematical concept / From Wikipedia, the free encyclopedia

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In mathematics, the **domain of a function** is the set of inputs accepted by the function. It is sometimes denoted by $\operatorname {dom} (f)$ or $\operatorname {dom} f$, where *f* is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".^{[1]}

More precisely, given a function $f\colon X\to Y$, 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 $f\colon X\to Y$, 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 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 $f\colon X\to Y$ to $A$, where $A\subseteq X$, is written as $\left.f\right|_{A}\colon A\to Y$.