# On mapping notation [duplicate]

So I’m in the midst of coming to grips with mapping notation and just require some clarity. Is there anything wrong with writing $$x\mapsto\frac{1}{x}$$ Because from my understanding, I understand that for this mapping to describe a function, we would need to specify that $x\neq0$, which we could then write as $$f:x\mapsto\frac{1}{x},x\neq0$$but assuming we’re not looking to describe the mapping of a function, is there anything wrong with the first expression?

The reason for my question is because I know that a mapping and a function are not the same thing, but we can use one to describe the other. Any responses are appreciated.

• What do you mean by "mapping" and "function"? The precise difference between these words is somewhat a matter of opinion. – Yakov Shklarov Jan 21 '18 at 5:16
• In my opinion, you don’t need to specify that $x\neq0$ because the natural domain already excludes $0$. – gen-ℤ ready to perish Jan 21 '18 at 6:24

I would write it as follows: $$\mathbb{R}\setminus\{0\}\to\mathbb{R}\;\colon\; x\mapsto \frac{1}{x}.$$ If you want, you can name your function if you use it later. Say you want to call it $f$. Then you would write $$f\colon \mathbb{R}\setminus\{0\}\to\mathbb{R}\;\colon\; x\mapsto \frac{1}{x}.$$ I believe this notation dates back at least to N. Bourbaki.

• Beautiful answer, right to the point and accurate. – gen-ℤ ready to perish Jan 21 '18 at 6:23
• @ChaseRyanTaylor thanks! I am a fan of this notation, actually. – max_zorn Jan 21 '18 at 7:57

Yes there are two things wrong with the first expression.

First of all the name of the function is missing.

If you are doing mathematics with a function , you better name the the function first.

For instance, if you take derivative of your function , how would you write it?

The second missing component in the first notation was the domain of your function.

note Well, the function $f(x) = 1/x$ is not defined at $x=0$ so you better mention that we do not allow $x$ to be zero.

• It's not always necessary to name a function. Sometimes it is appropriate to mention a specific function in passing, for example as in the phrase "as an example consider the map $x \mapsto 1/x$". As for the domain, often it can be deduced from context and doesn't have to be explicitly spelled out. – Yakov Shklarov Jan 21 '18 at 5:21
• @YakovShklarov I understand your point., but it does make it easier if you have a name for your function. For example the factorial function n! has a name and it is easy to work with it. – Mohammad Riazi-Kermani Jan 21 '18 at 5:39
• The whole point was to use mapping notation to express just a map, not a function; what you’re instructing is to turn the original expression into the map of a function, which as I’ve stated is not what I’m trying to do. – joshuaheckroodt Jan 21 '18 at 12:50
• @joshuaheckroodt As far as I know, the words function and mapping are synonymous and you may choose one or the other. – Mohammad Riazi-Kermani Jan 21 '18 at 14:32