Suppose the function f satisfies the functional equation f(a,b)=f(a+b,b-a) for all real a and b, and define g by g(x)=f(4^x,0). Show that g is periodic.
Let's just first play with f(a,b):
f(a,b) = f(a+b,b-a)
= f(2b,-2a)
= f(2b-2a,-2a-2b)
= f(-4a,-4b)
So after four compositions of the definition of f() we have f(a,b)=f(-4a,-4b). Then applying this composition to itself we get f(a,b)=f(16a,16b).
This last relation is very easy to apply to g(). Then g(x)=f(4^x,0)=f(4^(x+2),0)=g(x+2). Taking the leftmost and rightmost sides we get g(x)=g(x+2). Thus g() is periodic with period 2.
Solution
