Is the domain of the function $f(x)=1/x$ restricted or not?

Question

The domain of a function is the set of values that we are allowed to plug into our function so this means inputs over which our function is defined. So when it comes to the function $f(x)=1/x$, I thought that the domain would be all real numbers except 0 But apparently, that is the restricted domain but why?! ( about 5 or 6 sites that I think are quite good when it comes to mathematics said this )

I already know a bit about the restriction of a function which is a function obtained by choosing a smaller domain $A$ for the original function $f$. So domain $A$ is a subset of our general domain.

So why is the domain of that function restricted?

What book/text did you find that usage in? to me a restricted domain is when you take something that originally is defined on more than what you want, and simply state your altered smaller domain. It is completely up to whoever is using the restricted domain, and cannot be deduced just from a formula. — coffeemath, Oct 20 '23 at 14:09
Formally a function is defined by 3 things. The Domain, the Range and a law of a mapping. — dmtri, Oct 20 '23 at 14:10
@dmtri I believe you meant "domain, codomain, and the law of the map" [because range is determined by the domain and the law] Or maybe you use range for codomain and image for range? — coffeemath, Oct 20 '23 at 14:12
coffeemath I have the same mindset as you. These are the links to the sites that led to my confusion https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://courses.lumenlearning.com/intermediatealgebra/chapter/restricting-the-domain/&ved=2ahUKEwi925Sj6ISCAxVTg_0HHYJsBvQQFnoECBUQBQ&usg=AOvVaw0cci4PdONtwzpl3BcWcxHG — Credence, Oct 20 '23 at 14:16
And there is this one https://flexbooks.ck12.org/cbook/ck-12-precalculus-concepts-2.0/section/1.4/primary/lesson/domain-and-range-pcalc/ — Credence, Oct 20 '23 at 14:19
@Credence I scanned that and it seems poorly written to me. Doidn't scn your last one but I would not take these too seriously. There are multiple nmes in use for the set Y in f:X -> Y, both codomain and range are used, and multiple names for the set of outputs, both image and range, making "range" itself a bad choice since some use it one way and some the other. — coffeemath, Oct 20 '23 at 14:20
@coffeemath, No I meant Range. For example the function $f(x) =1/x$ with range $\mathbb(R^} {0} ) $ and range $\mathbb{R} $ it is not the same with that having range the raels except 0. The first is surjective and the second not surjective. — dmtri, Oct 20 '23 at 14:22
@dmtri what you are calling "range" is called "codomain" by some writers. Even more a source of confusion is that some writers use "range" to denote the set of values f(x) for x in the domain. These are mere definitions but I've seen both used. — coffeemath, Oct 20 '23 at 14:28
@Coffeemath sorry for interrupting your conversation with dmtri but can you explain your last sentences " Even more a source of confusion is that some writers use "range" to denote the set of values f(x) for x in the domain. " I always thought that that was the correct usage — Credence, Oct 20 '23 at 14:35
For some basic information about writing mathematics at this site see, e.g., here, here, here and here. — Henrik supports the community, Oct 20 '23 at 14:42
Well, defining the function on $\mathbb R\setminus {0}$ is a restriction compared to defining it on $\mathbb C\setminus{0}$. — Henrik supports the community, Oct 20 '23 at 14:49
@Henrik supports the community, I thought about this too but in the sites I visited they wrote it was because we are not considering zero in our domain — Credence, Oct 20 '23 at 14:57
@dmtri Both image and range are used by different writers for the set often called f(X) where X is the domain. And the term codomain is used by some to mean Y in f:X->Y, as also some use the term range for Y. — coffeemath, Oct 20 '23 at 14:58
@coffeemath, Thanks for the clarification. Sometimes these things look also obscure to me. — dmtri, Oct 20 '23 at 15:04
@dmtri To me this multiple usage of terminology is unfortunate, but in well written work is cleared up by the author's own explicit definitions of terms used. There are other cases of this in mathematics, I guess inevitable as writers from different times or places discuss similar notions. — coffeemath, Oct 20 '23 at 16:06
@user21820 oh dear, I am very sorry. I have a very important physics exam coming up this week and I completely forgot about my problems on domains and ... And yes, I understood most parts but there are parts that I need to think about it more ☕ if I have any questions can I ask you later ?! — Credence, Nov 01 '23 at 12:25
@Credence: Of course! If it's specific to my answer, you can ask in comments below it. If it's not entirely related, please ping me and I'll request a moderator to grant you access to my chat-room. — user21820, Nov 01 '23 at 12:34
Hello! I happened to come here again. Do you want to ask any more questions? — user21820, Jan 16 '24 at 07:14
@user21820 Oh yes, I have a ton of questions ☕ sorry about the delay in my response, my 1 and a half months of exams are finally over, and I requested permission to enter the chat room you mentioned. Thank you so much again ☕ — Credence, Jan 16 '24 at 19:12

score 2 · Accepted Answer · answered Oct 21 '23 at 07:11

The domain of a function is the set of values that we are allowed to plug into our function so this means inputs over which our function is defined.

This is correct.

So when it comes to the function $f(x)=1/x$

This is wrong.

"$f(x)=1/x$" is not a function; it is an equation with an undefined symbols "$f$" and "$x$".
"$f(x)$" is not a function either; it is an expression with undefined symbols "$f$" and "$x$".
"$1/x$" is not a function either; it is an expression with undefined symbol "$x$".

The only thing that can be a function in "$f(x)=1/x$" is "$f$", but you need to define it first.

Defining a function $f$ requires that you specify its domain and exactly how each input is mapped to an output. You cannot write down an equation and claim to define a function, despite the countless horrible high-school textbooks that do that.

I thought that the domain would be all real numbers except 0.

In high-school, you may have to cry inside and realize that when a test paper writes "What is the domain of the function $f(x)=1/x$?" they are wrong and it should have been written as "What is the largest $D ⊆ ℝ$ such that $1/x$ is well-defined for every $x ∈ D$?". And the answer would then be $ℝ{∖}\{0\}$. It is still not the domain of any function in the question, because there is simply no function defined in the question.

But apparently, that is the restricted domain but why?!

This is just nonsense. There is no such thing as a "restricted domain" of a function. Furthermore, it's worse for anyone to say that "$ℝ{∖}\{0\}$" is the "restricted domain of the function $f(x)=1/x$", because not only is there no function defined, but also there cannot be a function $f : D→ℝ$ such that $D ⊆ ℝ$ and $f(x)=1/x$ for each $x∈D$!

about 5 or 6 sites that I think are quite good when it comes to mathematics said this

Even Math SE is not that good when it comes to mathematical pedagogy. Not to say other websites, most of which are far worse.

I already know a bit about the restriction of a function which is a function obtained by choosing a smaller domain $A$ for the original function $f$. So domain $A$ is a subset of our general domain.

I do not know what you mean by "general domain". Again, there is no such thing. However, you are correct that if you have any function $f : D→S$ and any $A ⊆ D$, then you can construct the function $f{↾}A : A→S$.

So why is the domain of that function restricted?

Whoever said this is not a mathematician.

Is the domain of the function $f(x)=1/x$ restricted or not?

1 Answers1