Noel's function
Level
pending
Let f(x) be a real-valued function such that f(x) + 2f((x+1)/(x-1)) = 1/x for all values of x except -1, 0 and 1. If f(4) = a/b where a and b are positive coprime integers, what is a-b?
The answer is 773.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
f(x) + 2f((x+1)/(x-1)) = 1/x ---------(1) Replacing x with (x+1)/(x-1), we get f((x+1)/(x-1)) + 2f(-1/x) = (1-x)/(1+x) ----------(2) Doing the same for (2), we have f(-1/x) + 2f((x-1)/(x+1)) = -x ----------(3) And doing the same for (3), we finally get f((x-1)/(x+1)) + 2f(x) = (x+1)/(x-1) --------(4) Now, (1) - 2 * (2)+ 4 * (3) - 8 * (4): -15f(x) = 1/x - 2(1-x)/(1+x) - 4x - 8(x+1)/(x-1) Hence 15f(x) =-1/x + 2(1-x)/(1+x) + 4x + 8(x+1)/(x-1) Substituting x=4 into this equation we get 15f(4) = -1/4 - 6/5 + 16 + 40/3 = 1673/60 Thus, f(4) = 1673/900 where a = 1673 and b= 900. Therefore a-b = 773.