# 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).

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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• Okay now I’m also confused. (@5)
Why don’t we just say its codomain is {1}?