f(xy) = f(x) + f(y) is true for all

Consider the following statements :

Statement I The function f : R $$\to$$ R such that f(x) = x³ for all x$$\in$$R is one-one.

Statement II f(a) = f(b) $$\Rightarrow$$ a = b for all a, b $$\in$$R, if the function f is one-one.

Which one of the following is correct in respect of the above statements ?

If $$f({x_1}) - f({x_2}) = f\left( {{{{x_1} - {x_2}} \over {1 - {x_1}{x_2}}}} \right)$$ for x₁, x₂ $$\in$$($$-$$1, 1), then what is f(x) equal to ?

What is the range of the function $$y = {{{x^2}} \over {1 + {x^2}}}$$ where x$$\in$$R ?