Image (of a function)

The image \(\operatorname{im}(f)\) of a func­tion \(f : X \to Y\) is the set of all pos­si­ble out­puts of \(f\), which is a sub­set of \(Y\). Us­ing set builder no­ta­tion, \(\operatorname{im}(f) = \{f(x) \mid x \in X\}.\)

Vi­su­al­iz­ing a func­tion as a map that takes ev­ery point in an in­put set to one point in an out­put set, the image is the set of all places where \(f\)-ar­rows land (pic­tured as the yel­low sub­set of \(Y\) in the image be­low).

Domain, Codomain, and Image

The image of a func­tion is not to be con­fused with the codomain, which is the type of out­put that the func­tion pro­duces. For ex­am­ple, con­sider the Ack­er­mann func­tion, which is a very fast-grow­ing (and difficult to com­pute) func­tion. When some­one asks what sort of thing the Ack­er­mann func­tion pro­duces, the nat­u­ral an­swer is not “some­thing from a sparse and hard-to-calcu­late set of num­bers that I can’t tell you off the top of my head”; the nat­u­ral an­swer is “it out­puts a num­ber.” In this case, the codomain is “num­ber”, while the image is the sparse and hard-to-calcu­late sub­set of num­bers. For more on this dis­tinc­tion, see the page on codomain vs image.

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