Can the codomain of a function not include image points that are in the range. F

Question

Can the codomain of a function not include image points that are in the range. For instance, f: R---> [0,Infinity) where f(x)==x. See the codomain is only from 0 to infinity, but the range can include -1 for example and other negative numbers. So I guess this means the range is bigger than the codomain. Can that ever happen?

Explanation / Answer

NO, THIS CANNOT HAPPEN it cannot ever happen that a function has a range which is not a subset of its co domain. When the function is defined then its defined as a subset of the cross product of the domain and the codomain so if f:A -> B then f is defined as a subset of AxB where every element of A occurs exactly once. So it cannot happen that range contains a point which is not in the co domain. I know there are a lot of answers which say its not a problem, but it is if it were not so then why would anyone need a co domain at all?? Check the definition of the function. This definition is not a valid function. message me if you have any doubts.

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