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The image records the values that a map actually attains. A function is declared with a codomain, but the codomain may contain many elements that the function never reaches; the image separates the declared target from the realized target. This distinction is basic in [Set](/page/Set) theory, [Function](/page/Function) theory, [Linear Map](/page/Linear%20Map) theory, and algebra, where the image of a structure-preserving map often becomes a subobject of the codomain.

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For example, a map $f: A \to B$ may have codomain $B$ even when all values of $f$ lie in a smaller subset $C \subsetneq B$. The image is the canonical way to name that smaller subset without changing the original map. It is the target actually seen by the domain.

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## Definition

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The first question a function asks is not only where it is allowed to land, but where it lands after all inputs are used. The image answers this by collecting all output values produced by elements of the domain.

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[definition: Image of a Function] Let $A$ and $B$ be sets, and let $f: A \to B$ be a function. The image of $f$ is the subset of $B$ defined by \begin{align*} \operatorname{im} f = f(A) := \{f(a) : a \in A\}. \end{align*} [/definition]

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To use images in arguments about restricted domains, we also need a version that sends only a chosen subset forward. This refinement is essential whenever a proof tracks how a particular part of the source moves under a function.

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Many arguments need the image not only for the whole domain, but for a selected part of it. This lets a map transfer local or restricted information from a subset of the domain into the codomain.

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[definition: Image of a Subset] Let $A$ and $B$ be sets, let $f: A \to B$ be a function, and let $S \subset A$. The image of $S$ under $f$ is the subset of $B$ defined by \begin{align*} f(S) := \{f(s) : s \in S\}. \end{align*} [/definition]

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In algebra, this subset version becomes more powerful once the function preserves operations, because the forward image may inherit structure from the source. We need the following definition to isolate the linear case, where reached outputs become a vector subspace rather than only a subset.

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[definition: Image of a Linear Map] Let $U$ and $V$ be vector spaces over a field $k$, and let $T: U \to V$ be a linear map. The image of $T$ is \begin{align*} \operatorname{im} T := \{T(u) : u \in U\} \subset V. \end{align*} [/definition]

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For a general homomorphism, the important question is not only which elements are hit, but whether the hit elements still form a structure of the same kind inside the codomain. Naming this subset uniformly lets later statements compare it with kernels, quotients, and exactness without repeating separate language for groups, rings, modules, and vector spaces.

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[definition: Image of a Homomorphism] Let $X$ and $Y$ be algebraic structures of the same type, and let $\varphi: X \to Y$ be a homomorphism. The image of $\varphi$ is \begin{align*} \operatorname{im} \varphi := \{\varphi(x) : x \in X\} \subset Y. \end{align*} [/definition]

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The phrase "same type" means that $X$ and $Y$ might be groups, rings, modules over a fixed ring, vector spaces over a fixed field, or another category where homomorphisms have been defined. In each setting, the structural content of the image is supplied by the operations preserved by $\varphi$.

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## Equivalent Characterisations

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The definition says that the image is a set of values. A useful way to recognize the same subset is by membership: an element of the codomain belongs to the image exactly when it is hit by some input.

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[quotetheorem:9481]

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Once membership in the image is expressed as solvability of an equation, the next question is whether every point of the codomain is solvable in this sense. We need a separate definition for functions whose image fills the whole codomain.

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[definition: Surjective Function] Let $A$ and $B$ be sets, and let $f: A \to B$ be a function. The function $f$ is surjective if \begin{align*} \operatorname{im} f = B. \end{align*} [/definition]

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To compare forward movement with backward movement under a function, we also need the companion operation that starts with a subset of the codomain and returns the inputs that land there. This next definition gives the backward operation that behaves differently from images and is often the more stable tool.

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[definition: Preimage of a Subset] Let $A$ and $B$ be sets, let $f: A \to B$ be a function, and let $T \subset B$. The preimage of $T$ under $f$ is \begin{align*} f^{-1}(T) := \{a \in A : f(a) \in T\}. \end{align*} [/definition]

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