On mapping notation

Question

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.

Answer 1

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.

Answer 2

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.