For $x\ge y$, is the set of functions that satisfy the $\operatorname{f}(x)\gt \operatorname{f}(y)$ condition a vector space?
How should we approach this question before applying vector space axioms or how can we apply vector space axioms to determine that the function set is a vector space or not?
To start, remember that the space of functions (on $\mathbb{R}$ here) is a vector space (with addition and scalar multiplication working as you expect them to, 1 is the multiplicative identity, and f = 0 is the additive identity). The set of functions you're dealing with here is then a subset of this vector space. All that is left for you to do is to verify whether this subset is indeed a vector space (i.e. contains the additive identity and is closed under addition and scaling).