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 5 of 5: Determine the constant and verify
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In plain words

Consistency on the image fixes the remaining constant.

c=1,f(x)=1−x22c=1,\quad f(x)=1-\frac{x^2}{2}
Detailed analysis

For x x in the image, the first-step formula and the global formula must agree; this forces c=1 c=1. Substitution verifies that f(x)=1−x2/2 f(x)=1-x^2/2 satisfies the equation, so it is the unique solution.