• Support PF! Buy your school textbooks, materials and every day products Here!

Question concerning functions

  • #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
 

Answers and Replies

  • #2
RUber
Homework Helper
1,687
344
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.
 

Related Threads on Question concerning functions

Replies
2
Views
787
Replies
2
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
615
Replies
1
Views
682
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
1
Views
956
Replies
3
Views
623
Replies
1
Views
782
Replies
3
Views
2K
Top