A functional equation problem: $f(f(x))=x^2-x+1$

Asked Jan 24 '22 at 06:17

Active Jan 24 '22 at 17:33

Viewed 207 times

I have found this functional equation problem in here https://gonitzoggo.com/archive/problem/423/english This problem states that

for a function $f:\mathbb{R}\to\mathbb{R}$, $$f(f(x))=x^2-x+1$$ Upon these given conditions, find the sum $f(1971)$+$f(50)$+$f(2021)$.

I've made some progress by setting $$f(f(f(x)))=f(f(f(x)))$$ $$\implies f(x^2-x+1)=f(x)^2-f(x)+1$$ Plugging in $x=1$, after solving the equation $f(1)=f(1)^2-f(1)+1$, we find that $f(1)=1$.

As the equation's discriminant, $x^2-x+1$ is negative, so $f(x)\ge0 \;\forall x\in\mathbb{R}$.

Also, I've found that $f(0)$ is either $0$ or $1$.

Can anyone help me to find the full solution for the problem?

edited Jan 24 '22 at 17:33

Mohsen Shahriari

7,071

asked Jan 24 '22 at 06:17

Saad Junior

7

Does this answer your question? Do there exist functions $f$ such that $f(f(x))=x^2-x+1$ for every $x$? – Jan 24 '22 at 06:30
1

Related: "Do there exist functions $f$ such that $f(f(x))=x^2-x+1$ for every $x$?", "How much does $(f \circ f)(x)=x^2 - x + 1$ determine $f$?" and "If $f(f(x))=x^2-x+1$, what is $f(0)$?". Apparently, the problem of proving $ f ( 0 ) = 1 $ has originally appeared as Problem 5 of Baltic Way 2011 contest. The same question posted recently: "Functional Equation - $f(f(x)) = x^2 - x + 1$". – Mohsen Shahriari Jan 24 '22 at 17:29

A functional equation problem: $f(f(x))=x^2-x+1$

0 Answers0