Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"
- Anindya Biswas
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Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, \[f(f(f(x)+y))=f(x+y)+f(x)+y\]
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
- Anindya Biswas
- Posts:264
- Joined:Fri Oct 02, 2020 8:51 pm
- Location:Magura, Bangladesh
- Contact:
Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"
"If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is."
— John von Neumann
— John von Neumann
Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"
$P(x,f(y))\Rightarrow f(f(f(x)+f(y)))=f(x+f(y))+f(x)+f(y)$Anindya Biswas wrote: ↑Thu May 13, 2021 12:12 amFind all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, \[f(f(f(x)+y))=f(x+y)+f(x)+y\]
$\Rightarrow f(x+f(y))=f(y+f(x))$
$\Rightarrow f(x+y)+f(y)+x=f(x+y)+f(x)+y$
$\Rightarrow f(x)=x+f(0)$
Plugging this into the main equation wee see that no such function exists.$\blacksquare$