Given a map $p: E\to B$ and some point $b\in B$, what are some ways to show that a set $F$ is the preimage set $p^{-1}(\{b\})$? So far, it seems $F\subseteq p^{-1}(\{b\})$ can be shown by taking points in $F$ and applying $p$, but how do I know there isn't some point in $E\setminus F$ that maps to $b$?
For context, I'm trying to prove that the Hopf map $\eta:S^3\to S^2$ defined by \begin{align*} \eta(x) = x\cdot k\cdot x^{-1} \end{align*} (via quaternions) is surjective and that the fiber is $S^1$. If $x=(w,x,y,z)$, then we have \begin{align*} \eta(x)=(0,2wy+2xz,2yz-2wx,w^2+z^2-x^2-y^2). \end{align*} I'm fairly new to quaternions, so any insight/advice will be appreciated.
Unless I'm mistaken, the coordinate form you have written down is slightly off. The third entry should be $2yz-2wx$.
Fiber bundles are among other things quotient maps, so, when trying to characterize them, it's natural to ask if there is an equivalence relation or group action which produces the quotient. This is particularly useful here, since $S^3$, viewed as the unit quaternions, is a Lie group. Consider the subset of unit quaternions given by $$ \Gamma=\{\cos(\theta)+k\sin(\theta):\theta\in\mathbb{R}\} $$ $\Gamma$ is clearly isomorphic to $S^1$ (as a group as well as a manifold), and is a Lie subgroup of $S^3$. I claim that the Hopf map is equivalent to the quotient map $\pi:S^3\to S^3/\Gamma$.
From your starting point, this means showing $\eta(q_1)=\eta(q_2)$ iff $q_1=q_2\gamma$ for some $\gamma\in\Gamma$. From this it follows almost immediately that the fibers of $\eta$ are circles.
As for surjectivity, it suffices to find an element in the preimage of an arbitrary point. Since conjugation by $x$ is a rotation in the $2$-sphere of unit imaginary quaternions, you can find a preimage of an arbitrary imaginary unit quaternion $v$ by finding a quaternion representing a rotation that takes $k$ to $v$.
If the quaternion approach isn't very intuitive, it's also common to define the Hopf bundle using complex numbers (this amounts to a change of coordinates). Wikipedia gives an outline of this construction.