whether the function is injective surjective or both

Question

Let $f: \mathbb R^2\to\mathbb R$ be defined by $f(x,y)=x^2+y^3,$ then is it injective bijective or both?

I try this: if $x_1^2+y_1^3=x_2^2+y_2^3$, it doesn't mean $x_2=x_1, y_2=y_1$. So it is not injective.

Am I correct?

Answer 1

The function is a surjection (not an injection):

Proof that it's not an injection: $f(1,0)=f(-1,0)=1$.

Proof it's a surjection: $f(0,\sqrt[3]x)=x$ for all $x\in\mathbb R$.