Is $x^2+x+1 (mod 2)$ a one-way function?
A one-way function is a function from $\{0,1\}^*$ to $\{0,1\}^*$ which satisfies certain properties. Your function only accepts a single bit as input, so in particular isn't a one-way function.



We don't know whether one-way functions exist, though many people conjecture that they do. Proving that one-way functions exist is harder than proving $\mathsf{P} \neq \mathsf{NP}$, that is, if one-way functions exist then $\mathsf{P} \neq \mathsf{NP}$. Therefore proving that a given function is one-way is expected to be very difficult, though some candidates do exist.