A couple of questions about Lyapunov functions

Question

Consider the dynamical equation $\dot{x} = f(x)$, where $f:\mathbb{R}_+ \rightarrow \mathbb{R}_+$.

1. Is $V(x) = x^{1+\alpha}$, where $\alpha \in (0,1)$ a Lyapunov function?
2. Since $x(t) \ge 0$ for all $t$, would a function $V(y) \not\in \mathbb{R}_+$ for $y \not\in \mathbb{R}_+$ still be a Lyapunov function?