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I would write,
Let $f,g:\mathbb{C}\rightarrow\mathbb{C}$ be defined by
$$f(x)=\frac{x^3+2x^2+\sin(x)}{x^5-x^3+\cos(x)+5}$$
and
$$g(x)=\frac{x^3-5x+\log(x)}{x^7+3x^2+10}.$$
Consider $f\circ g$.
Giving complicated things names (and using English/your native language to explain them) is a good thing.
Oddly enough, there was some discussion about this sort of notation earlier: here on meta and here on MO. I tend to favor the more verbose choices of notation given above, because I think they are less likely to cause confusion for the reader.
The person who just voted and left a comment on the poll seems to have missed the part that said, "This poll has now run its course".
I don't understand the fuss being made by people in this discussion. I have no objection to writing $f(x)$ as a function. I frequently do this in the context of $f$ being a polynomial. In other contexts I don't. Just as I sometimes write $\int f(x) d\mu(x)$ when integrating a function $f$ with respect to a measure $\mu$, and other times I just write $\int f d\mu.$ It depends on the context, and what I plan to do next with the formula.
I am also one of the people who voted in Andrew Stacy's poll, who has no objection to writing $f(x)$ to denote a function.
In the same vein, writing $f(x)/x$ (say) to denote the quotient of one function by another seems perfectly fine to me, in the appropriate context. I can see that there are times when it may be a source of confusion (because of issues of the precise domain), and then one would avoid it, but there are lots of other contexts in which it would be perfectly okay (the most common, for me, being if $f(x)$ was a rational function, or more generally an analytic function, of $x$).
I agree that I wouldn't normally use the composition symbol $\circ$ in the way that Anixx did in his post at the top of this thread, just because I think it looks a little strange, but it's meaning is certainly clear, and I don't see what is so objectionable about it.
Incidentally, I am a pure mathematician.
Anixx, a function of two real variables is a function of one variable from R^2. For example, if I had f:R->R^2 and g:R^2->R, there's nothing wrong with writing g \circ f.
You are too focused on writing the functions with their arguments "on display" - I think you would do well to think of functions more categorically, with the domain and codomain built into the definition of the function, and without there necessarily being "arguments" to work with because the objects you are working with are not necessarily sets. In this understanding, the function f:R->R defined by f(x)=0 and the function g:R->C defined by g(x)=0 are different functions, for example. Furthermore, it is part of the definitions of a category that one can only compose functions f and g when we have f\in Hom(A,B) and g\in Hom(B,C), where A, B, and C are objects of your category.
I very much agree with Alexander Woo, in that I think of a function as an object in and of itself, and not as a mere relation between independent and dependent variables.
As to your second comment, it just depends on what I feel like emphasizing. It's obvious that a subscript is just another place to put an argument, i.e. defining the function $f:X->Z$ by $f(x)=B_x(a)$, where {B_x:Y->Z}_{x\in X} is a collection of functions indexed by X and $a\in Y$, is equivalent to defining $f:x->Z$ by $f(x)=B(x,a)$, where $B:X x Y ->Z$ and $a\in Y$, and $B(x,y)=B_x(y)$ where $B_x$ are the functions from before. If for some reason I felt like keeping the setup of an indexed family of functions, it would be perfectly fine to write B_x for a specific x\in X, as that is the name of a specific function. If I converted the indexed family of functions into a big function B (again, they are equivalent), then I would be fine with just talking about B, without requiring myself to write in an x in the first argument of B. However, I expect that people would for the most part understand what I meant if I never explicitly mentioned the change but still wrote B without a subscript.
Dear Zev,
I think that one should be careful making categorical statements of the form "You are too focused on writing the functions with their argments "on display"". Anixx uses a certain system of notation which is completely reasonable, possibly slightly idiosyncratic when considered in one or two of its details, but which has much in common with systems of notation used by many other mathematicians and scientists who use mathematics as a tool. Furthermore, my impression based on Anixx's posting history is that her/his topics of interest are all closely related to classical analysis. I don't see that there is any particular need "to think of functions more categorically" in that context.
Incidentally, how do you denote a sequence? A sequence is a function from the natural numbers to the real numbers (say), and almost everyone I know denotes such a thing $a_n$, not just $a$, i.e. they put the variable $n$ on display in their notation. This is convenient for many purposes, even if it is occasionally bothersome (in that certain expressions and constructions become more convoluted when forced into this notational convention). Furthermore, most people write $a_{n_i}$ for a subsequence, rather than something more functorial such as $a \circ j$ where $j$ is a monotone injection from the natural numbers to itself. The notation $f(x)$ is simiarly traditional in many circles (even if it is not as universal as the sequence notation $a_n$). Like sequence notation, it serves some purposes, and can be bothersome in others.
Finally, I do often denote the identity function from an algebraically closed field (say) to itself by $x$. This is very standard in algebraic geometry, where $x$ stands for both a polynomial variable and the (identity) function on the field of coefficients that it induces. I don't think this reflects a lack of functorial thinking; it is just a particular notation tradition.
Regards,
Matthew
Emerton, my impression so far was that Anixx has not agreed that the categorical way of thinking of functions has any value - I was attempting to identify the point on which he is being stubborn, or at least the point on which our disagreement lies, so that we could effectively discuss it. The "you" was referring only to Anixx. I didn't mean to come across as claiming that writing f(x) (or the expression for f(x), if there is one) for the function f was universally bad - I don't think that at all - I just mean that Anixx is for whatever reason very intent on writing it that way, that this stubbornness is causing some confusion for him, and that considering an alternative point of view would be good for him, personally. For example, I expect that Anixx would not have had concerns about how to write the composition of functions with more than 1 variable if he had considered composition from a domain/codomain/categorical standpoint. I do agree that Anixx's questions are on a subject where the categorical approach is not required, but again, I feel that it would be helpful for Anixx to at least appreciate what Alexander Woo eloquently describes as "the reification of the function".
I stand by my answer in the case of the particular setup that Anixx proposed, namely a family of functions indexed by an arbitrary set. I completely agree that it is customary, and quite often useful, to write a sequence as a_n, not a:N->R. However, I would point out that it's common, at least in my experience, to use "a" as a name for the sequence (at least when it's not being used for the limit of the sequence), i.e. $a={a_n}_{n\in N}$, which I consider an acceptable middle ground between only ever writing a_n's (and ignoring the sequence as an object itself), and defining "a" to be a function a:N->R.
I will attempt to think of a better way of phrasing my previous post, but I've got to go to sleep for now.
@Emerton- I have to disagree with you on reasonableness. The issue isn't that Anixx uses the notation f(z), but that s/he uses it in a way which doesn't make sense. Writing f(z) \circ g(z) for f(g(z)) is not a reasonable thing to do, and it was notational choices like that which spawned this thread.
Dear Ben,
I agree that $f(z) \circ g(z)$ is not good notation, and so "completely reasonable" was too strong a claim. The point remains, though, that there is nothing wrong with writing $f(x)$ to denote a function, or $\sin(x)/x$ for that matter.
Best wishes,
Matt
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