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 1 of 5: Evaluate the function on its image
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In plain words

Choose the input equal to a value of the function.

c=f(0),f(a)=c+1−a22 (a∈f(R))c=f(0),\quad f(a)=\frac{c+1-a^2}{2}\ (a\in f(\mathbb R))
Detailed analysis

Let c=f(0) c=f(0) and let a=f(y) a=f(y) be in the image. Substituting x=a x=a gives c=2f(a)+a2−1 c=2f(a)+a^2-1, hence f(a)=(c+1−a2)/2 f(a)=(c+1-a^2)/2.