Does this special function exist?

Question

Is it possible to find a non-trivial function $f(x_1,x_2)$ that has two parameters $x_1$ and $x_2$.

This function should satisfy $f(x_1,x_2) = f(\frac{x_1}{1+r},x_2 +r)$, for any non-negative $r$.

For instance, if $f(x_1,x_2) = x_1 x_2$, then $f(x_1,x_2) = f(x_1 r, \frac{x_2}{r})$.

Thanks.

Answer 1

It needs to form a group, where addition of $r1$ then $r2$ is the same as adding $r1+r2$. But then $$f(x1,x2)=f(\frac{x1}{1+r1+r2},x2+r1+r2), or\\ f(x1,x2)=f\left(\frac{x1}{1+r1},x2+r1\right)=f\left(\frac{x1}{(1+r1)(1+r2)},x2+r1+r2\right)$$ so $f$ has to be independent of $x1$.