For reference, I am in Introduction to Abstract Math, and I have taken Calculus 1, Discrete Math, and Introduction to Linear Algebra.

I am trying to prove or disprove that $g : ( \mathbb{R}^+ \times \mathbb{R}) \to \mathbb{R}^2, \:\:g(x,y)=(x^2-y^2,xy)$ is one to one.

I know the definition of a one-to-one function is that a function $f : A \to B$ is one-to-one if for all $a\_1, a\_2 \in A$, whenever $f(a\_1) = f(a\_2)$, then $a\_1=a\_2$.

Due to the domain of $g$, I know that $x>0$, and I also have proven that $g$ is not onto with the counterexample of there being no $g(x,y)$ such that $g(x,y)=(0,0)$. I also have values for $x^2$ and $-y^2$, which are $x^2=\frac{a+\sqrt{a^2+4b^2}}{2}$ and $-y^2=\frac{a-\sqrt{a^2+4b^2}}{2}$. However, I am unsure of where to go from here.