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I'm having a little trouble understanding the difference between the codomain and the range of a function.

I'm reading the Wikipedia article on Codomain (I can't post the URL), but it doesn't make sense.

I understand what they are saying and it still doesn't make any sense. How are the f and g functions not the same? How can you arbitrarily define a codomain for a function that is a specific superset of the range?

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