Problem 5

Let

S

be the set of all real numbers greater than

-1

. Find all functions

f:S\to S

such that

f(x+f(y)+xf(y))=y+f(x)+yf(x)

for all

x,y\in S

, and

\frac{f(x)}{x}

is strictly increasing on each of the intervals

-1<x<0

and

0<x

Step 1 of 4: Propagate a fixed point

Previous step Next step

In plain words

Substituting a fixed point twice produces another fixed point.

f(a)=a\Longrightarrow f(2a+a^2)=2a+a^2

Detailed analysis

If $f(a)=a$ , put $x=y=a$ in the functional equation. Both sides then show that $b=2a+a^2$ is fixed: $f(b)=b$ . For $-1<a<0$ , we have $-1<b<a$ ; for $a>0$ , we have $b>a$ .