A question about FE

For discussing Olympiad Level Algebra (and Inequality) problems
Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm
A question about FE

Unread post by Asif Hossain » Wed May 05, 2021 2:04 pm

Does $f(xf(x))=xf(x) \Rightarrow f(x)=x$ $\forall x \in \mathbb{R}$??(I don't think so :| as if $f(x)=1/x$ we would get a constant value each time or may be other weird function may satify this inversive property making it constant each time but idk )
Hmm..Hammer...Treat everything as nail

User avatar
Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: A question about FE

Unread post by Mehrab4226 » Wed May 05, 2021 9:09 pm

Asif Hossain wrote:
Wed May 05, 2021 2:04 pm
Does $f(xf(x))=xf(x) \Rightarrow f(x)=x$ $\forall x \in \mathbb{R}$??(I don't think so :| as if $f(x)=1/x$ we would get a constant value each time or may be other weird function may satify this inversive property making it constant each time but idk )
I think the problem with $f(x)=\frac{1}{x}$ is that its domain is not $\mathbb{R}$, it is $\mathbb{R} \backslash \{0\}$. But the function should have a domain of the whole set of real numbers. This is what I think, but I may be wrong too.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: A question about FE

Unread post by Asif Hossain » Wed May 05, 2021 10:19 pm

Mehrab4226 wrote:
Wed May 05, 2021 9:09 pm
Asif Hossain wrote:
Wed May 05, 2021 2:04 pm
Does $f(xf(x))=xf(x) \Rightarrow f(x)=x$ $\forall x \in \mathbb{R}$??(I don't think so :| as if $f(x)=1/x$ we would get a constant value each time or may be other weird function may satify this inversive property making it constant each time but idk )
I think the problem with $f(x)=\frac{1}{x}$ is that its domain is not $\mathbb{R}$, it is $\mathbb{R} \backslash \{0\}$. But the function should have a domain of the whole set of real numbers. This is what I think, but I may be wrong too.
Doesn't $x=0 \Rightarrow f(0)=0$? then the domain $\mathbb{R} \backslash \{0\}$ shouldn't mess around. :|
Hmm..Hammer...Treat everything as nail

User avatar
Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: A question about FE

Unread post by Mehrab4226 » Thu May 06, 2021 4:02 am

Asif Hossain wrote:
Wed May 05, 2021 10:19 pm
Mehrab4226 wrote:
Wed May 05, 2021 9:09 pm
Asif Hossain wrote:
Wed May 05, 2021 2:04 pm
Does $f(xf(x))=xf(x) \Rightarrow f(x)=x$ $\forall x \in \mathbb{R}$??(I don't think so :| as if $f(x)=1/x$ we would get a constant value each time or may be other weird function may satify this inversive property making it constant each time but idk )
I think the problem with $f(x)=\frac{1}{x}$ is that its domain is not $\mathbb{R}$, it is $\mathbb{R} \backslash \{0\}$. But the function should have a domain of the whole set of real numbers. This is what I think, but I may be wrong too.
Doesn't $x=0 \Rightarrow f(0)=0$? then the domain $\mathbb{R} \backslash \{0\}$ shouldn't mess around. :|
SO I guess your function is $f(x)=\frac{1}{x}$ if $x \neq 0$, and $f(0)=0$ if $x=0$. I guess then it works.
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: A question about FE

Unread post by Asif Hossain » Thu May 06, 2021 12:37 pm

Mehrab4226 wrote:
Thu May 06, 2021 4:02 am
Asif Hossain wrote:
Wed May 05, 2021 10:19 pm
Mehrab4226 wrote:
Wed May 05, 2021 9:09 pm

I think the problem with $f(x)=\frac{1}{x}$ is that its domain is not $\mathbb{R}$, it is $\mathbb{R} \backslash \{0\}$. But the function should have a domain of the whole set of real numbers. This is what I think, but I may be wrong too.
Doesn't $x=0 \Rightarrow f(0)=0$? then the domain $\mathbb{R} \backslash \{0\}$ shouldn't mess around. :|
SO I guess your function is $f(x)=\frac{1}{x}$ if $x \neq 0$, and $f(0)=0$ if $x=0$. I guess then it works.
Actually the main question is does $f(xf(x))=xf(x) \Rightarrow f(x)=x\ \ \forall x \in \mathbb{R}$? :roll:
Hmm..Hammer...Treat everything as nail

User avatar
Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: A question about FE

Unread post by Mehrab4226 » Thu May 06, 2021 2:05 pm

Asif Hossain wrote:
Thu May 06, 2021 12:37 pm
Mehrab4226 wrote:
Thu May 06, 2021 4:02 am
Asif Hossain wrote:
Wed May 05, 2021 10:19 pm


Doesn't $x=0 \Rightarrow f(0)=0$? then the domain $\mathbb{R} \backslash \{0\}$ shouldn't mess around. :|
SO I guess your function is $f(x)=\frac{1}{x}$ if $x \neq 0$, and $f(0)=0$ if $x=0$. I guess then it works.
Actually the main question is does $f(xf(x))=xf(x) \Rightarrow f(x)=x\ \ \forall x \in \mathbb{R}$? :roll:
Probably not
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: A question about FE

Unread post by Asif Hossain » Thu May 06, 2021 10:38 pm

Mehrab4226 wrote:
Thu May 06, 2021 2:05 pm
Asif Hossain wrote:
Thu May 06, 2021 12:37 pm
Mehrab4226 wrote:
Thu May 06, 2021 4:02 am

SO I guess your function is $f(x)=\frac{1}{x}$ if $x \neq 0$, and $f(0)=0$ if $x=0$. I guess then it works.
Actually the main question is does $f(xf(x))=xf(x) \Rightarrow f(x)=x\ \ \forall x \in \mathbb{R}$? :roll:
Probably not
why? i have just shown example of such functions but can it proven?(except this existential contradiction)
Hmm..Hammer...Treat everything as nail

Dustan
Posts:71
Joined:Mon Aug 17, 2020 10:02 pm

Re: A question about FE

Unread post by Dustan » Fri May 07, 2021 6:19 pm

Main problem konta?

User avatar
Mehrab4226
Posts:230
Joined:Sat Jan 11, 2020 1:38 pm
Location:Dhaka, Bangladesh

Re: A question about FE

Unread post by Mehrab4226 » Fri May 07, 2021 7:41 pm

Dustan wrote:
Fri May 07, 2021 6:19 pm
Main problem konta?
Does $f(xf(x))=xf(x)$ imply $f(x)=x \forall x \in \mathbb{R}$
The Mathematician does not study math because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful.
-Henri Poincaré

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: A question about FE

Unread post by Asif Hossain » Tue May 11, 2021 11:00 pm

Bumpty Bumpty Bump
Hmm..Hammer...Treat everything as nail

Post Reply