Re: Can we find the function?

On Oct 31, 2:54 pm, Thomas Nordhaus <thnord2...@xxxxxxxx> wrote:
polymedes schrieb:

On Oct 31, 1:55 pm, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
polymedes <polyme...@xxxxxxxxx> wrote:
Let (f o f)(x) = -2x + 3
Can we find the original function f ?
How about f(x) = Sqrt(-2) x + 1 - Sqrt(-2) ?

(Of course, I'm not saying that is the only such function.)

David

yes, if we assume that it is a polynomial function, then it is easy to
solve the a(ax+b)+b=-2x+3. But, is it? Or better, is there any other
function with the same property (i.e. (f o f)(x) = -2x + 3)?

Not a real function f. It can't have a fixed point x0 because g = f o f
would have x0 as a fixed point too and g'(x0) = (f'(x0))^2 >=0 (chain
rule) leads to a contradiction. So either f(x) > x for all x or f(x) < x
for all x. So the orbit of a point under f is either increasing or
decreasing. But that doesn't work since the orbits of a point x0 under g
are alternating except for x0 = 1.

--
Thomas Nordhaus

Let the function be a complex function. f(x) = i*sqrt(2) x + 3/
(i*sqrt(2)+1).

.