Page 1 of 1
Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"
Posted: Thu May 13, 2021 12:12 am
by Anindya Biswas
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, \[f(f(f(x)+y))=f(x+y)+f(x)+y\]
Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"
Posted: Thu May 13, 2021 2:36 pm
by Dustan
Easy!
Ans
Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"
Posted: Thu May 13, 2021 3:38 pm
by Anindya Biswas
Dustan wrote: ↑Thu May 13, 2021 2:36 pm
Easy!
Ans
Re: Problem - 03 - National Math Camp 2021 Mock Exam - "Functional equation, but not functioning well!"
Posted: Fri May 14, 2021 11:21 am
by ~Aurn0b~
Anindya Biswas wrote: ↑Thu May 13, 2021 12:12 am
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, \[f(f(f(x)+y))=f(x+y)+f(x)+y\]
$P(x,f(y))\Rightarrow f(f(f(x)+f(y)))=f(x+f(y))+f(x)+f(y)$
$\Rightarrow f(x+f(y))=f(y+f(x))$
$\Rightarrow f(x+y)+f(y)+x=f(x+y)+f(x)+y$
$\Rightarrow f(x)=x+f(0)$
Plugging this into the main equation wee see that no such function exists.$\blacksquare$
Ans