KingOfActing
lukewarm mess
So in my boredom these holidays I have thought up the following functional equation:
 = \frac{1}{f(x)})
With the only restriction being that it is continuous on
.
I first looked at constant solutions, which gave me = \pm 1)
Then linear solutions were all = \pm x)
And in general, any polynomial solution is
for some natural n. Just through inspection it becomes clear that this works for any real n.
I then went on to try rational functions, and after solving for general linear/linear and quadratic/quadratic, I noticed the pattern that the general rational function which is a solution to this functional equation is of the form
. This solution also includes all previous solutions as well.
I've also found:
$ is a solution, then so is $ \frac{1}{g(x)}\\$If $g(x)$ and $h(x)$ are both solutions, then so is $g(x)h(x)\\f(1) = \pm 1\\f(-1) = \pm 1)
Is there any more general form for the solution than the one I've found? Any transcendental functions/non-elementary functions that satisfy the equation?
With the only restriction being that it is continuous on
I first looked at constant solutions, which gave me
Then linear solutions were all
And in general, any polynomial solution is
I then went on to try rational functions, and after solving for general linear/linear and quadratic/quadratic, I noticed the pattern that the general rational function which is a solution to this functional equation is of the form
I've also found:
Is there any more general form for the solution than the one I've found? Any transcendental functions/non-elementary functions that satisfy the equation?