Problem : Prove that the map $f:\mathbb{R}^2\to \mathbb{R}$ defined by :
$f(x,y)=y^3+xy^2+x+y$
is a quotient map.
My attempt : Surjecctive part is clear. Now at first I note that the map $g:\mathbb{R}\to \mathbb{R}$ defind by $g(x)=ax+b$ is continuous. Next I note that the maps $g_1 , g_2:\mathbb{R}\to \mathbb{R}$ defined by $g_1(x)=x^3+x$ and $g_2(x)=x^2+1$ are continuous. Now can I in any way use the fact that composition of quotient maps is a quotient map to prove the result. Any other technique to solve the problem will also be appreciated.
Observe that $f$ is a submersion ($Df(u,v)=(v^2+1,3v^2+2uv+1)$ and $v^2+1>0$). And sumbmersion is an open map. And $f$ is also surjective. So a surjective open map is a quotient map.