MathLabs

Problem 5

Let SS be the set of all real numbers greater than −1-1. Find all functions f:S→Sf:S\to S such that f(x+f(y)+xf(y))=y+f(x)+yf(x)f(x+f(y)+xf(y))=y+f(x)+yf(x) for all x,y∈Sx,y\in S, and f(x)x\frac{f(x)}{x} is strictly increasing on each of the intervals −1<x<0-1<x<0 and 0<x0<x.
Step 4 of 4: Verify the candidate
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In plain words

The fractional-linear map preserves the domain and has the required ratio monotonicity.

f(x)=−xx+1f(x)=-\frac{x}{x+1}
Detailed analysis

For x>−1x>-1, f(x)+1=1x+1>0f(x)+1=\frac1{x+1}>0, so f(x)∈Sf(x)\in S. Also f(x)x=−1x+1\frac{f(x)}x=-\frac1{x+1} has derivative 1(x+1)2>0\frac1{(x+1)^2}>0 on both required intervals. Substitution into the equation verifies it, so this is the unique solution.