Suppose $\eta: \mathbb{A}^1 \to X$ is a morphism of varieties over $\mathbb{C}$, such that for each $b \in \mathbb{A}^1$, $\eta(b)$ only depends on $b^2$, i.e. $\eta(b) = \eta(-b)$. Does it follow that $\eta$ factors over the following branched covering? \begin{align*} f\colon\mathbb{A}^1 & \to \mathbb{A}^1 \\ b & \mapsto b^2 \end{align*} That is, is there a map $\xi: \mathbb{A}^1 \to X$ such that $\eta = \xi \circ f$?
Set-theoretically (at least on closed points) it is clear. Even in the analytic contex, I could locally choose the branch of the logarithm, define a square root, and then patch the local functions together. But how can one formally show this in the context of algebraic geometry?