Mnemonic: in means into. So we are going into a codomain that is large enough so that we can have a different image for every input.

Surjective function

Mnemonic: sur means over. So we are going over the codomain, and covering it entirely.

Domain of a function

Vs: image: the codomain is the set that the function might reach.

The image is the exact set that it actually reaches.

E.g. the function:

f (x) = x^{2}

could have:

Note that the definition of the codomain is somewhat arbitrary, e.g.

x^{2}

could as well technically have codomain:

R ⋃ R^{2}

even though it will obviously never reach any value in

R^{2}

The exact image is in general therefore harder to characterize.

A function where the domain is the same as the codomain.

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