I have the problem :
Find all functions f:R→R that satisfy
f(x)+3f(x−1x)=7x
for all nonzero x. Note that the domain of f is all real numbers. Make sure you define your function everywhere!
Hint: If you substitute (x−1)/x in for x, then you get a new equation that also involves f(x−1x)Then do this substitution again and you may get another useful equation to consider.
I think this is related to the fact that f(x)=x−1x is cyclic with order 3, but I don't know how to use this to find the answer.
StackExchange...huh...I answered some questions there about computer science. I didn't know they also had a math webpage.
In the proof, 0 is restricted because the denominator of a fraction cannot be 0 (cannot divide by 0). If we set x to 1, then the part 3f(x−1x) cancels out and turns into 0 because 3f(0)=0. Also, if you look around in the proof, you will find x and 1−x many times as the denominator, which cannot equal 0. Simply stated, 0 and 1 are restricted, AKA not in the domain of the function.
Hope this helps,
- PartialMathematician