# f(x) + 2x = f(f(x))

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### William Elliot

May 24, 2009, 4:01:38 AM5/24/09
to
Are there any solutions, R into R, for
f(x) + 2x = f^2(x)

other than f(x) = -x and f(x) = 2x?

### alainv...@gmail.com

May 24, 2009, 7:01:29 AM5/24/09
to

Bonjour,

Three notices:
1°) sum(b(i)f^i(x)) = 0 ,b(i)constant, i integer
Be r(j) the roots of sum(b(i)*a^i) = 0 ,
possible solutions on C f(x)= r(j)x .

2°) One fixed point x=0 ,

3°)a formal relation:
from 2x = f^2(x) -f(x)
x = {(f-I)/2} o f(x) ,I identity
Thence (f-I)/2} = f^ -1

Alain

### Patrick Coilland

May 24, 2009, 8:30:34 AM5/24/09
to
William Elliot a �crit :

> Are there any solutions, R into R, for
> f(x) + 2x = f^2(x)
>
> other than f(x) = -x and f(x) = 2x?

-x and 2x are the only two continuous solutions of f(x)+2x=f(f(x)).

But infinitely many non continuous solutions exist.

### Robert Israel

May 24, 2009, 3:02:32 PM5/24/09
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Patrick Coilland <pcoi...@pcc.fr> writes:

E.g. take any subset A of R such that (t -> -t) maps A to itself and
(t -> 2t) maps R \ A into itself, and let f(x) = -x for x in A and
2x for x in R \ A.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

### Martin Musatov

May 24, 2009, 3:08:43 PM5/24/09
to

Hi Robert,
I would rather have my name associated with this triviality:P=NP as
was proven:
MMM