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

if x = y, then
f(y) + 8f(1/y) = 63y

If x = 1/y, then
8f(y) + f(1/y) =  63/y

This is a pair of linear, simultaneous equations which can be solved for f(y), by multiplying the 2nd equation by 8 and subtracting.

-63f(y) = 63y - 63*8/y

f(y) = (8/y) - y

There is no other solution.

Posted by Steve Herman on 2006-10-18 23:58:28