Question 14
Find all real-valued functions $f$ defined on the set of all non-zero real numbers such that:
(i) $f(1)=1$,
(ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x, y, x+y$,
(iii) $(x+y) f(x+y)=x y f(x) f(y)$ for all non-zero $x, y, x+y$.
(i) $f(1)=1$,
(ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x, y, x+y$,
(iii) $(x+y) f(x+y)=x y f(x) f(y)$ for all non-zero $x, y, x+y$.