Determine all functions `f : R rarr R` satisfying `f(x-f(y)) = f(f(y)) + xf(y) + f(x) - 1 AA x, y in R`

Answer

`f(x)=1-(x^(2))/(2)`

Explanation

Given `f(x-f(y))=f(f(y))+xf(y)+f(x)-1 " (1) " `
Putting `x=f(y)=0,` we get `f(0)=f(0)+0+f(0)-1,` i.e.,
`f(0)=1 " (2)" `
Again, putting `x=f(y)= lambda` in (1), we get
`f(0)=f(lambda)+lambda^(2)+f(lambda)-1`
or `1=2f(lambda)+lambda^(2)-1 " [From (2)]" `
`f(lambda)=(2-lambda^(2))/(2)=1-(lambda^(2))/(2)`
Hence, `f(x)=1-(x^(2))/(2).`