Answer: -f(x) can be odd but cannot be even.
Explanation:
we are given that for any p<q
f(p)<f(q)
this clearly implies that f is an increasing function.
Now we know that if f is an increasing function then -f is always an decreasing function and vice-versa.
so here -f(x) will be an decreasing function.
Let us consider a example f(x)=x then f(x) is clearly an increasing function.
and -f(x)= -x is an decreasing function. also it is an odd function but not an even function.