I am using the Hopf Map $f: \mathbb{S}^3 \rightarrow \mathbb{S}^2$ where $f(a,b,c,d) = (2(ab+cd),2(ad-cb),(a^2+c^2)-(b^2+d^2))$
My question is
1.) How can I prove surjectivity of the map? I am aware of the primary definition of surjectivity but I am unsure how to apply in this case.
I tried using the trivial method of finding an inverse but the method is very complicated.
2.) How can I show that a point that the preimage of $q \in \mathbb{S}^2$ is a circle in $\mathbb{S}^3$?
I realize that the solution is dependent on finding the inverse function as hinted in the first part, but since the function itself was complicated to find, hence I had no progress in this part.