Proof that a relation is in FP
To show that $R \in \mathsf{FP}$ you need to show that there is a deterministic poly-time algorithm $A$ that takes $x$ as input and outputs a $y$ such that $(x,y) \in R$.
A possible choice of $A$ is the algorithm that outputs $\varepsilon$ (the empty word) regardless of $x$. Indeed, $(x, \varepsilon) \in R$.