# Function / Map notation?

Please forgive my ignorance, if I've phrased my question improperly. I'm not sure what the appropriate terminology is; that's the basis of my question. So, I'm not sure if I'm even remotely close in my description of this notation:

I'm wondering what is the proper name for the following notation. Is it function notation? Or part of set notation? None of the mathematics courses I've taken at my college (College Algebra through Calculus II) has used this notation, but I've begun to encounter it in some of the calculus & analysis textbooks I've looked at, and I see it used here and elsewhere on the web.

The other part of my question is, where can I learn more about the following notation (and similar/related notation)? It seems like notation like this should be covered in some course, but I haven't encountered on yet which did cover it:

$$f : \mathbb{R} \to \mathbb{R} \quad \text{ or } \quad f : \mathbb{Z} \to \mathbb{R}$$

I understand this is a form of function notation, but I wasn't sure what to call it, and thus what would make a good google query. When I googled "function notation," of course, the results I got back were about the familiar $y=f(x)$ notation.

• As for the name, I don't know. If you haven't already found this wikipedia page, it's a good place to start and become familiar with it.
– Em.
Dec 15, 2015 at 3:26
• Yeah, I did find that page and actually read through the section about notation before posting. It has an example of the type of notation I'm talking about, but it doesn't label it. Admittedly, I didn't read the entire (quite long) page. But the notation section seemed most relevant. Interestingly, it did mention the notation I'm asking about in the context of domain and codomain (vs. range, which is the terminology used in all the mathematics courses I've taken at my college). Dec 15, 2015 at 6:05
• It would be nice if one time they introduced it. I think you are expected to pick it up when you first see it without being taught it. There isn't much to it at least. Dec 15, 2015 at 6:13
• @tommytwoeyes: FYI, here's one difference between codomain and range. If we write $f(x) = x^2$ where $f: \mathbb{R} \mapsto \mathbb{R}$, then the codomain is not equal to the range, since the range (all values the function actually takes on, as opposed to those it "could" take on) is $\mathbb{R}^{\ge 0}$. You may already know this; I just think it's cool! Dec 15, 2015 at 6:20
• @fleablood Yes, I couldn't agree more. They should work notation into the beginning of a pre-calculus or calculus course, or require students to take a mathematical thinking/induction course which includes it. Dec 16, 2015 at 7:48

$f:A\to B$ means that $f$ is a function, that the domain of $f$ is $A$, and that $f(a)\in B$ for every $a\in A.$.... For example, if you see " for every $f:A\to B$ " it means " for every $f$ such that $f:A\to B$ ". In other words " for every function $f$, with domain $A$ , that maps $A$ into $B$ ".
• I can see why people like that notation, then. The first time I saw it, without knowing what it meant of course, I thought it looked like some funky Haskell or Lua syntax. But now that you've clarified it, it does evidently convey a lot more information than the $f(x)$ notation I'm used to (which requires actual work to find the range). ;) Dec 16, 2015 at 7:52
• I've seen it all over the place, at every level. It saves a lot of words. Since the notion of a function is ubiquitous, there are quite a few varying notations. I always thought that is unfortunate that we write $f(x)$, not $(x)f,$ because ,an expression like $f(g(x))$ has to read from right to left to be understood. Fine in Hebrew,not in English. Dec 16, 2015 at 8:47
• I see your point. But I like the current notation, especially for combination functions. I was a web developer way before I began studying mathematics, somewhat paradoxically, so notation like $f(g(x))$ is completely natural to me. Dec 21, 2015 at 1:51