If a function `f: R ->R` be such that `f(x-f(y)) = f(f(y) )+xf(y)+f(x) -1 AA x , y in R` then `f(2)=`

Question

If a function `f: R ->R` be such that `f(x-f(y)) = f(f(y) )+xf(y)+f(x) -1 AA x , y in R` then `f(2)=`
A. 1
B. 3
C. `-1`
D. none of these

Answer 1

Correct Answer - C
We have,
`f(x-f(y))=f(f(y))+xy(dy)+f(x)-2" (1)"`
Put `x=f(y)=0`
Then `f(0)=f(0)+0+f(0)-1`
`therefore" "f(0)=1`
Putting `f(y)=x`
`therefore" "f(0)=f(x)+x^(2)+f(x)-1`
Hence, `f(x)=1-(x^(2))/(2)`