Functions – Is the Domain of f(x)=1/x Restricted?

functions

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?

Best Answer

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.

  1. "$f(x)=1/x$" is not a function; it is an equation with an undefined symbols "$f$" and "$x$".
  2. "$f(x)$" is not a function either; it is an expression with undefined symbols "$f$" and "$x$".
  3. "$1/x$" 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.

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