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Think of "order preserving" as increasing: if a<b then f(a) < f(b).
The thing about algorithms is that they always work... but may not be the most efficient way to approach a problem.
We've done more in the narrative of the 4th edition to address this, and other opportunities, to shortcut established algorithms provided the underlying assumptions allow one to do so.
For example, the solution to sqrt(x) > 2 is true for all x >= 0, but, squaring both sides results in just x > 4.
