# Image (mathematics)

## Set of the values of a function / From Wikipedia, the free encyclopedia

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In mathematics, for a function $f:X\to Y$, the **image** of an input value $x$ is the single output value produced by $f$ when passed $x$. The **preimage** of an output value $y$ is the set of input values that produce $y$.

More generally, evaluating $f$ at each element of a given subset $A$ of its domain $X$ produces a set, called the "**image** of $A$ under (or through) $f$". Similarly, the **inverse image** (or **preimage**) of a given subset $B$ of the codomain $Y$ is the set of all elements of $X$ that map to a member of $B.$

The **image** of the function $f$ is the set of all output values it may produce, that is, the image of $X$. The **preimage** of $f$, that is, the preimage of $Y$ under $f$, always equals $X$ (the domain of $f$); therefore, the former notion is rarely used.

Image and inverse image may also be defined for general binary relations, not just functions.