Let f:R→R satisfy

f(a)≠0 for some a in R

f(xf(y))=yf(x) for all x,y in R
Prove that f(x)=f(x) for all x in R.
To figure out what possible functions f could be, input a/f(a) for x and a for y: (for nonzero a)
f((a/f(a))*f(a))=a*f(a/f(a)) which means f(a)/a=f(a/f(a))
If we set b=a/f(a), then it reduces to f(b)=1/b. For any points on f(a)=a, it reduces to f(1)=1. For other points, if a point exists on f(a)=ka, then f(k)=1/k and f(1/k)=k.

Posted by Gamer
on 20061229 14:11:48 