TITLE: Are surjective functions a pointless concept?
QUESTION [11 upvotes]: What is the point of the concept of surjective or onto functions if you can just restrict the co-domain of your function to its image? Injectivity or 1-to-1ness is actually the defining property of bijections (sometimes called 1-to-1 correspondences), while all the question of surjectivity does is derail the argument into checking whether the image equals the co-domain.
Am I wrong in thinking this? What am I missing? Only situation where this concept could be marginally useful that I could think of would be some function for which it is easier to find an element of the co-domain for which no pre-image exists than to actually find the image itself.

REPLY [2 votes]: What is the point of the concept of surjective or onto functions if you can just restrict the co-domain of your function to its image?

I read this statement as questioning the point of using larger sets only when we know the image of a map, although it logically doesn't say anything about when we don't know the image; that is, I assume that OP sees the importance of codomains (which may be larger than the image) in defining the concept of function, whether or not their image is easy to determine. Having said that, in cases where the image of the function is easy to determine, it has been determined only because you still (implicitly or explicitly) have first assumed the type of outputs you want your function to have. The set of these is what we call the codomain, whether or not we can explicitly specify the range of values of the function. That is, suppose you have an expression $f(x).$ Then to completely determine its meaning as a function, you must know the class of objects to which $x$ belongs, and you must also know the class of objects to which $f(x)$ belongs -- this is important, for changing this class changes (i.e., limits or extends the range of) the operation itself, so that not even the range is yet fixed, until we have fixed this class of objects we want $f(x)$ to be in. Clearly, then, you can see that whether you like it or not you cannot escape the concept of codomains if you want to completely define the meaning of function at all, talk less of if you want to talk about ranges. Thus, it is basic, necessary to the concept of function. The image of a mapping lies in some set, whether we explicitly admit this or not -- this is the codomain. But some functions take all values in their (assigned) codomain -- clearly these have a property that not all functions with the same codomain have (recall that a function is not specified until the codomain is -- I think this is something you're yet to fully appreciate). We like it when we encounter such singular behavior. They are therefore called surjective, or similar terms.

Injectivity or 1-to-1ness is actually the defining property of bijections (sometimes called 1-to-1correspondences), while all the question of surjectivity does is derail the argument into checking whether the image equals the co-domain.

As explained above, you can't coherently and definitely think of a function without implicitly or otherwise making assumptions about its domain of action and its codomain of products. Thus, it appears that injectivity is only well-defined relative to having first fixed a codomain. Of course injectivity is an interesting behavior in itself, hence our also singling it out. But this doesn't detract a bit from the interestingness of surjectivity. They're different concepts. Now, you say that injectivity is the defining property of bijections, and this is true. Recall that a bijection is a function that must have a domain both ways; thus, if we do not know the range of a map, although it's injective, it's pretty much useless for most anything. However, since the business of finding ranges is not altogether neat, that is why you usually find the statement that if a function is injective and is surjective, then it is bijective. I don't know the book you're using, but unless they state it as a definition (which then should not be a problem -- a definition doesn't always claim to exhaust a class of objects; it only defines any class it finds interesting or convenient to deal with), then you need not read this as an if-and-only-if statement; that is, in such case they're not necessarily saying that surjectivity is also necessary for a map to be bijective. Thus, I now hope you see that the question of the surjectivity of an injective function does not derail anything, at least if you really want to use functions for something, and not merely think generally and abstractly about them.

Am I wrong in thinking this? What am I missing? Only situation where this concept could be marginally useful that I could think of would be some function for which it is easier to find an element of the co-domain for which no pre-image exists than to actually find the image itself.

In summary then, you are wrong in your first quote above in the sense that you can't think of ranges without thinking about codomains first; indeed, you can't conceive of a function uniquely without also specifiying its codomain, among other requirements. In your second paragraph as quoted above, the question of injectivity in the most general sense depends only on injectivity, but in practise it's easier to single out surjective functions that are also injective; this automatically guarantees their bijection in an explicit manner, ready for use. This however doesn't mean that only surjective functions can be possibly bijective, except you take the preceding statement as a definition, which is not a problem anyway.