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 3 of 5: Represent an arbitrary input
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In plain words

Use the image-difference property in the original equation.

x=a−b⇒f(x)=f(b)+ab+f(a)−1x=a-b\Rightarrow f(x)=f(b)+ab+f(a)-1
Detailed analysis

For arbitrary x x , choose image values a,b a,b with x=a−b x=a-b . Applying the equation with first variable a a and with f(y)=b f(y)=b gives f(x)=f(b)+ab+f(a)−1 f(x)=f(b)+ab+f(a)-1.