Question 102
Let $f$ be function defined from the set $\{(x, y): x, y \text{ reals}, x y \neq 0\}$ into the set of all positive real numbers such that
(i) $\quad f(x y, z)=f(x, z) f(y, z)$, for all $x, y \neq 0$;
(ii) $\quad f(x, y z)=f(x, y) f(x, z)$, for all $x, y \neq 0$;
(iii) $\quad f(x, 1-x)=1$, for all $x \neq 0,1$.
Prove that
(a) $\quad f(x, x)=f(x,-x)=1$, for all $x \neq 0$;
(b) $\quad f(x, y) f(y, x)=1$, for all $x, y \neq 0$.
(i) $\quad f(x y, z)=f(x, z) f(y, z)$, for all $x, y \neq 0$;
(ii) $\quad f(x, y z)=f(x, y) f(x, z)$, for all $x, y \neq 0$;
(iii) $\quad f(x, 1-x)=1$, for all $x \neq 0,1$.
Prove that
(a) $\quad f(x, x)=f(x,-x)=1$, for all $x \neq 0$;
(b) $\quad f(x, y) f(y, x)=1$, for all $x, y \neq 0$.