Let
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
be the set of all real numbers strictly greater than −1. Find all functions
![f: S \to S](/media/m/8/5/4/85486f3f9575951f5e47ae88e6dd1281.png)
satisfying the two conditions:
(a)
![f(x + f(y) + xf(y)) = y + f(x) + yf(x)](/media/m/f/b/3/fb3b0a152e52c846ccf74544aad7a266.png)
for all
![x, y](/media/m/2/7/9/279a699b10f7b70e7160f4aaaa89e453.png)
in
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
;
(b)
![\frac {f(x)}{x}](/media/m/c/f/d/cfd8aa3bb9217f771ec5f9e80e8be387.png)
is strictly increasing on each of the two intervals
![- 1 < x < 0](/media/m/3/8/3/38374b03f6a34e1048da7b7a1f51889c.png)
and
![0 < x](/media/m/9/6/9/969d1bcfe72fef87f457da48bf1dcd71.png)
.
%V0
Let $S$ be the set of all real numbers strictly greater than −1. Find all functions $f: S \to S$ satisfying the two conditions:
(a) $f(x + f(y) + xf(y)) = y + f(x) + yf(x)$ for all $x, y$ in $S$;
(b) $\frac {f(x)}{x}$ is strictly increasing on each of the two intervals $- 1 < x < 0$ and $0 < x$.