Image (mathematics)

Algebraic structure → Group theory Group theory
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In mathematics, for a function ${\displaystyle f:X\to Y}$, the image of an input value ${\displaystyle x}$ is the single output value produced by ${\displaystyle f}$ when passed ${\displaystyle x}$. The preimage of an output value ${\displaystyle y}$ is the set of input values that produce ${\displaystyle y}$.

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

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

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