Solve F(x) for the given domain. Include all of your work in your final answer. Submit your solution. F(x) = x 2 + 2 F(x2) =

Question

I relally do no understand this concept.

Answer 1

F(x)=x^2+2F(x^2)=x^2+2(x^4+2F(x^4))=x^2+2x^4+4F(x^4)=x^2+2x^4+4(x^8+2F(x^8))=x^2+2x^4+4x^8+8F(x^8)=

x^2+2x^4+4x^8+8(x^16+2F(x^16))=... Well you can see what's happening. We have a series:

F(x)=x^2+2x^4+2^2x^8+2^3x^16+2^4x^32+2^5x^64+...

So each term increases the power of 2 by 1 and doubles the power of x. If x<1, the series converges, but if x>1 it diverges to infinity. If x=0, F(x)=0 and if x is between -1 and zero it converges to the same value as the opposite positive value, so F(x)=F(-x). Therefore the domain that converges is -1<x<1. Outside that range it diverges to infinity.