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 3 of 4: Apply the equation on the diagonal
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In plain words

Every input of this form is fixed, so it must equal the unique fixed point 00.

f(x+f(x)+xf(x))=x+f(x)+xf(x)f(x+f(x)+xf(x))=x+f(x)+xf(x)
Detailed analysis

Set x=yx=y in the functional equation. It gives f(k)=kf(k)=k for k=x+f(x)+xf(x)k=x+f(x)+xf(x). Since 00 is the only fixed point, k=0k=0 for every x∈Sx\in S, hence x+f(x)+xf(x)=0x+f(x)+xf(x)=0 and f(x)=−xx+1f(x)=-\frac{x}{x+1}.