determine all functions f(x) multiply with f(y)=f(x)+f(y)+xy-1

You need to determine all functions which are in the
relation f(x)*f(y) = f(x)+f(y)+xy-1

Put x=y=1 =>
f(1)*f(1) = f(1)+f(1)+1-1 => `f^2(1) =
2f(1)`

Subtract 2f(1) and then factor
f(1):

`f^2(1) -2f(1) = 0`  => `f(1)*(f(1) - 2) = 0`
=> f(1)=0

f(1) - 2 = 0 => f(1) =
2

Put f(1) = 0 => if y = 1 and x `in`  R, then the
relation  f(x)*f(y) = f(x)+f(y)+xy-1 suffers a
transformation.

f(x)*f(1) =
f(x)+f(1)+x-1

0 = f(x)+0+x-1 => f(x) =
1-x

If f(1) = 2 => y=1, x `in` 
R

2f(x) =  f(x)+2+x-1

Subtract
f(x)=> f(x) = x + 1

ANSWER: The
functions that check the relation f(x)*f(y) = f(x)+f(y)+xy-1 are f(x) = 1-x and f(x) = x
+ 1.