This is my question:
Let $A=\{(x,y) \in \mathbb{R^2}| x+y\neq -1\}$ and $f: A\to \mathbb{R^2}$ such that, $$f(x,y) =\left(\frac{y}{1+x+y}, \frac{x}{1+x+y}\right) \ \ \ \text{for every }(x,y) \in A.$$ Which of the following is/are true? a) $f$ is injective, b) $f(A)= \mathbb{R^2}$
My attempt: taking $f(x,y) = (a,b)$ then we get,
$$\begin{cases} -ax+(1-a)y= a& \\ (1-b)x - by = b \end{cases}$$
After solving above system we get that, determinant of coefficient matrix is non-zero and hence system has unique solution for every $(a,b)$ and hence $f$ is injective. Is am I right? but I can't able to prove/discard $f$ is surjective or not? Please help me.
Hint for surjectivity: try to solve $\,f(x,y)=(0,1)\,$ for $\,x,y\,$.