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Looking for functions (Posted on 2006-10-18) Difficulty: 3 of 5
Define all functions that are defined in the set of real numbers excluding zero and satisfy the following:

f(x)+8f(1/x)-63x=0 , x≠0

No Solution Yet Submitted by atheron    
Rating: 3.2000 (5 votes)

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Something missing? | Comment 1 of 6
Inserting x=1 and x=-1 delivers f(1)=7 and f(-1)=-7. Now if f_1 is ANY (partial) function defined for x>=1 and satisfying f_1(1)=7, then it can be extended to 0<x<=1 using the "definition"

f(x) := 63x - 8f(1/x)

Likewise ANY partial function f_2 defined for x<=-1 and satisfying f_2(-1)=-7 can be extended to -1<=x<0 using the above equation. So it seems there are many, many functions satisfying the equation.

This leads me to believe that atheron implicitly assumes additional requirements such as continuity, differentiability, holomorphic, etc. that are not given in the problem statement.

Edited on October 18, 2006, 1:38 pm
  Posted by JLo on 2006-10-18 13:38:20

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