1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Question concerning functions

  1. Jan 30, 2015 #1
    • Member warned about posting with no template
    I'm trying to solve this problem from a highschool math competition:
    Find all functions f : R → R such that, f(f(x+y)-f(x-y))=xy, for all real x,y.
    Any ideas of how to approach it.
    I have found that f(0)=0, if x=y f(f(2x))=x^2
  2. jcsd
  3. Jan 30, 2015 #2


    User Avatar
    Homework Helper

    Some thoughts:
    I would note that the domain and range of the function must be all real numbers. This should eliminate many of the trig functions and exponentials.
    Based on what you have, it seems like f(x) might incorporate some improper exponent...
    Say...##f(x) = \frac{x^\sqrt {2} }{A}## where A scales 2 to 1 over two iterations.
    I am not 100% sure how you would expand this out for the sums, and I don't think that function is defined for all x,y in the real numbers.

    Another option,
    Think of the derivatives:
    ##\frac{\partial}{\partial x} f ( f( x+ y) - f(x-y) ) = \frac{\partial}{\partial x} xy ##
    ##f ' ( f( x+ y) - f(x-y) ) * (f'(x+y)-f'(x-y)) = y ##
    ##\frac{\partial}{\partial y} f ( f( x+ y) - f(x-y) ) = \frac{\partial}{\partial y} xy ##
    ##f ' ( f( x+ y) - f(x-y) ) * (f'(x+y)+f'(x-y)) = x ##
    And second derivatives are all zero.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted