This is problem $4$ from "Basic Algebraic Geometry I" page $4$.
Let $X$ be the curve defined by the equation $y^{2}=x^{2}+x^{3}$ and $f:\mathbb{A}^{1} \rightarrow X$ defined by $f(t)=(t^{2}-1,t(t^{2}-1))$. Prove $f^{*}$ maps the coordinate ring $k[X]$ isomorphically to the subring of the polynomial ring $k[t]$ consisting of polynomials $g(t)$ such that $g(1)=g(-1)$. (Assume that chark $\neq 2$).
Let $W$ be the above subring. I can see that the image of $f^{*}$ is contained in $W$, I don't see the reverse inclusion, why is this?, also why $f^{*}$ is injective?