Why not always refer to a functions image, what is the point in specifying some super set of that image and then naming that super set the functions "codomain"? What does explicitly defining a set that contains more values for which your function does not even output add to the definition of a function.
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Related to / somewhat of a duplicate of http://math.stackexchange.com/q/5480/148175, http://math.stackexchange.com/q/97390/148175, http://math.stackexchange.com/q/777563/148175 – complexist May 29 '14 at 04:36
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1Pragmatically, if I don't know the precise range but I still want to give a generic description of the values of the function the concept of a codomain is useful. That said, there is much more in the links above... – James S. Cook May 29 '14 at 04:44
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It is a way of identifying surjectivity (a function that is onto)...it is equivalent that the range (image) and codomain are the same – afedder May 29 '14 at 05:26
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1@afedder So we define the notion of a codomain, so we can define another notion of surjectivity. Why is that helpful? – Ethan Splaver May 29 '14 at 05:46
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2Suppose I want to discuss the function $f:(-1,1)\to \mathbb{R}$ given by $f(x) = x^7 \sin x + e^{\cos x}.$ I may want to do various things like find its derivative or its roots. I can try to do these things without going to lengthy trouble of finding its precise image, which is really irrelevant to me. Finding the range can be tiresome, or extremely difficult. The point of giving the codomain is to give some information about the values, and so we have a precise idea of what type of objects we are dealing with. – Ragib Zaman May 29 '14 at 07:43
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@Ethan this is not the sole purpose of defining a codomain, there are many, I was only citing one use. Another one is as just referenced above dealing with the complications of computing ranges of functions...sometimes this is very simple, but other times it is quite tedious and maybe no need. – afedder May 29 '14 at 08:02