Prove that the function $h(x,y)=(y^3, xy^2 + x + 1)$ is injective, where $h: \mathbb{R^2} \to \mathbb{R^2}$.
I am trying to understand the form/notation meaning of $h(x,y)=(y^3, xy^2 + x + 1)$. All my examples have been in the range of $\mathbb{R}$ instead of $\mathbb{R^2}$.
Surjectivity:
Take any $(a,b)\in \mathbb{R}^2$. You have to find such $(x,y)$ that $$y^3=a\;\;\;\;{\rm and}\;\;\;\; xy^2+x+1=b$$
And that is easy to do. Take $y= \sqrt[3]{a}$ and $x= {b-1\over y^2+1}$ and we are done.
Injectivity:
Say $f(x,y)=f(x',y')$ then $y^3=y'^3$ (thus $y=y'$ since the map $x\mapsto x^3$ is injective) and $$xy^2+x+1 = x'y'^2+x'+1$$
so $$ x(y^2+1) = x'(y^2+1) \Longrightarrow x=x'$$ and we are done.