Implication value (for x which isn't in a domain of antecedent) We define two functions: $f(x): \dfrac{1}{x} \ge 0$ $g(x): x \ge 0$ and we want to find out for which $x$ implication $f(x) \Rightarrow g(x)$ is true. Obviously for all $x>0$ it's true ($1\Rightarrow1$). Similary for all $x<0$ it's true ($0\Rightarrow0$). However when it comes to $0$ I've got a problem. On the one hand, we can say that $0$ doesn't belong to domain of function $f$. But on the other hand, we can say that it belongs to domain of function $g$. Moreover $g(0)$ is true, so somehow we don't care whether $f(0)$ is true or false because if consequent is true then whole implication $f(0) \Rightarrow g(0)$ will always be true.

Terminologically, we'd normally call $f$ and $g$ predicates and not functions. From a logical perspective, there's no way of talking about a partially defined predicate typically. So you do need to say whether $f(0)$ is true or not. This comes down to what we mean by $\frac{1}{x}$. Taking a relational view of that, we can view a (total) function as a special type of binary relation, and a partial function is a similar such relation, so $\frac{1}{x}\geq 0$ means $\exists y.r(x,y)\land y \geq 0$ where $r$ stands for the relation representing the partial function $x\mapsto\frac{1}{x}$, e.g. $r(x,y)\equiv (xy = 1)$. This formula is well-defined and false when $x = 0$. You could potentially make other choices, but ultimately you do need to have a predicate be fully defined for it to be meaningful.