Differentiability Problem

Question

Supposing we are given relation that $$f(xy + 1)= f(x).f(y) - f(y) - x +2$$ and also given that $$f(0)=1$$ for a differentiable function then is function one-one onto? I partially differentiated relation first wrt to $x$ then $y$ $$ f'(xy + 1)y= f'(x).f(y) - 1$$ $$f'(xy + 1)x= f(x).f'(y) - f'(y)$$ Equating them and then integrating them with respect to any one variable should give me function but how to take I integral here?

Answer 1

$$ f(xy + 1)= f(x).f(y) - f(y) - x +2 $$ also, therefore $$ f(yx+ 1)= f(y).f(x) - f(x) - y +2 $$ hence $\forall x,y$ $$ f(x)-x = f(y)-y $$