if f(x)f(1/x)=f(x)+f(1/x) what can u say about f(x)?

Question

can you prove that f(x) must be x^n +- 1
or give any other info about f(x)

f(x) is a polynomial

Answer 1

Let g(x) = f(1/x)
We are given that:
f*g = f+g

Consider:
(f-1)*(g-1) = f*g - (f+g) + 1 = 0 +1 = 1
Clearly f-1 = 1 / (g-1)
So if f(x) is a polynomial, then f-1 = x^n fits that relation.
g-1 = x^-n
So f(x) = x^n + 1
g(x) = f(1/x) = x^-n + 1

Answer 2

There are many solutions other then x^n+1.

As others have shown, if we define g(x) = f(x)-1 then the problem is equivalent to finding solutions of g(x) g(1/x) = 1. If we let x=1, this implies g(1) = +1 or -1. But you can assign values of g(x) for x > 1 arbitrarily, provided that g(x) is not zero, and then define g(x) for 0 < x < 1 by g(x) = 1/g(1/x). The resulting function is a solution. If you want a continuous solution (you didn't say so), then you must choose g(x) for x > 1 in such a way that lim g(x) = g(1) as x -> 1.

[Added later]
Oh, NOW you tell me f is a polynomial... In that case the only solution is g(x) = x^n, because x^n is the only polynomial satisfying g(x) g(1/x) =1 (although I don't have a short proof of this, I think you can prove it by induction over the degree of the polynomial).

Answer 3

Since 2*2=2+2 the result of the function must always be 2.
f(x)=x°+1

Answer 4

f(x)f(1/x)
= f(x)+f(1/x)
= f(x)[1+f(1/x)/f(x)]

f(1/x) = 1+f(1/x)/f(x)
f(x) = f(1/x)/[f(1/x)-1] = 1 + 1/[f(1/x)-1]
or
f(x)-1 = 1/[f(1/x)-1]
Let g(x) = f(x)-1
g(x) = 1/g(1/x)
Therefore, g(x) = x^n
f(x) = x^n+1
----------
Further discussion:
Think about all elementary functions. Which one other than x^n has the property of g(x)g(1/x) = 1?

Answer 5

There are some benchmarks floating around. in case you observed then you rather could fall ill on your abdomen and you may probable purely stick to Intel. then you rather could ask your self why AMD is popping out with Bulldozer in the 1st place. you may take convenience in the undeniable fact that the benchmarks are not one hundred% reliable. i think those benchmarks come from area of the former u.s. that became into the farming usa and food grant for the previous Soviet block. you will could wait with something persons... till the tip of the week on the main. i'm individually not paying for something till techpowerup, guru3d, and anandtech have a evaluation to work out.

Answer 6

Its Simple
f(x)f(1/x)
= f(x)+f(1/x)
= f(x)[1+f(1/x)/f(x)]

f(1/x) = 1+f(1/x)/f(x)
f(x) = f(1/x)/[f(1/x)-1] = 1 + 1/[f(1/x)-1]
or
f(x)-1 = 1/[f(1/x)-1]
Let g(x) = f(x)-1
g(x) = 1/g(1/x)
Therefore, g(x) = x^n
f(x) = x^n+1

Answer 7

Hey. I found this website when I was in Algebra. It is myalgebra.com. Its really cool. It solves your problems for you. I am not sure about those kind, but check it out. I am not lieing!