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Question Number 194688 by CrispyXYZ last updated on 13/Jul/23

$$\mathrm{find}\mathrm{all}\mathrm{function}f:\mathbb{R}\to \mathbb{R}\mathrm{such}\mathrm{that}\mathrm{\forall }x,y\in \mathbb{R},$$$$f(x-f\left(y\right))=f\left(f\left(y\right)\right)+xf\left(y\right)+f\left(x\right)-1.$$

Answered by Tinku Tara last updated on 13/Jul/23

$$putx=f\left(y\right)$$$$f\left(0\right)=f\left(x\right)+x^2+f\left(x\right)-1$$$$f\left(0\right)=c$$$$f\left(x\right)=\frac{1+c-x^2}{2}$$$$puty=0$$$$f(x-c)=f\left(c\right)+cx+f\left(x\right)-1$$$$\frac{1+c-{(x-c)}^2}{2}=\frac{1+c-c^2}{2}+cx+\frac{1+c-x^2}{2}-1$$$$\frac{1+c-x^2}{2}+cx-\frac{c^2}{2}=\frac{1+c-c^2}{2}+cx+\frac{1+c-x^2}{2}-1$$$$\frac{1+c}{2}-1=0\Rightarrow c=1$$$$f\left(x\right)=1-\frac{x^2}{2}$$

Commented by CrispyXYZ last updated on 14/Jul/23

$$\mathrm{Thank}\mathrm{you}!$$

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