MathLabs

Problem 6

Determine all functions f:R→R f:\mathbb{R}\to\mathbb{R} such that f(x−f(y))=f(f(y))+xf(y)+f(x)−1 f(x-f(y))=f(f(y))+xf(y)+f(x)-1 for all real numbers x,y x,y .
Step 4 of 5: Simplify the difference formula
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In plain words

The dependence on the chosen image representatives cancels.

f(x)=c−x22f(x)=c-\frac{x^2}{2}
Detailed analysis

Insert the image formula from the first step for both f(a) f(a) and f(b) f(b). Since x=a−b x=a-b , the result is f(x)=c−b2/2+ab−a2/2=c−x2/2 f(x)=c-b^2/2+ab-a^2/2=c-x^2/2 for every real x x .