[จูกัดเหลียง]

Day I Problems2: $f:R\rightarrow R$ Find all function $$f(xf(x)+f(y))=f(x)^2+y$$
take $x=0$ we get $$f(f(y))=f(0)^2+y$$ and we take $y=-f(0)^2,f(y)=k$
Then get $f(k)=0$ so we can take that $x=y=k$ therefore $f(0)=0$
if we take $y=0$ we get $$f(xf(x)+0)=f(x)^2+0\leftrightarrow f(xf(x))=f(x)^2...(i)$$
or we take $x=0$ get $$f(f(y))=y\rightarrow f(f(x))=x...(ii)$$
The equation $(i)$ we take $x=f(x)$ then the equation is the following $f(xf(x))=x^2$
so by $(ii)$ we conclude that $f(x)=x,-x$