In my calculus book it mentions that increasing functions maintain inequality relations and that's the reason you can apply $\exp$ and $\ln$ to two sides of an inequality to solve them. Is there some general classification for the types of functions that maintain inequality? For instance are they all 1 to 1 or have some other property in common?
In addition to m. k.'s answer, there is one very valuable criterion: If a function is continuous, it is monotonic if and only if it is injective (this is a consequence of the Intermediate Value Theorem). Therefore, a continuous function $f: [a,b] \to \mathbb R$ will maintain a strict inequality if and only if $f(a) < f(b)$ and it is injective.