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I want to express the simple idea that two functions f and g are related in the following way:

$$ f(x) > f(y) \implies g(x) > g(y) $$

Or the inverse relation:

$$ f(x) > f(y) \implies g(x) < g(y) $$

I was going to use the term "proportional" or "inversely proportional", but I understand that in mathematics that implies a linear (or inverse linear) relationship, which is not my case. Is there a better term to express this than something like "$f(x)$ increases as $g(x)$ increases/decreases"?

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    $\begingroup$ I don't know a standard term, but if I were to make one up, I'd choose something like "$f$ and $g$ increase (in)coherently". $\endgroup$ – user228113 Sep 5 '17 at 15:12
  • $\begingroup$ Isn't this the problem that big and little "oh" notation are meant to address? $\endgroup$ – Xander Henderson Sep 5 '17 at 15:39
  • $\begingroup$ @XanderHenderson Mmm, well I was thinking more of a term, but in any case, I don't really want to say either $f \in \mathcal{O}(g)$ or $g \in \mathcal{O}(f)$, since none of them actually "bounds" the other... also I couldn't express the inverse relationship like that, I think. $\endgroup$ – jdehesa Sep 5 '17 at 16:00
  • $\begingroup$ If we're making up terms, "$f$ and $g$ are co-ordered". $\endgroup$ – Eric Towers Sep 5 '17 at 16:05
  • $\begingroup$ This has nothing to do with proportionality, nor with order of magnitudes, just with order. This question is a linguistic and semantic challenge indeed. $\endgroup$ – Christian Blatter Sep 5 '17 at 16:13
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Stick with your ideas.
f increases as g increases.
f increases as g decreases.

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  • $\begingroup$ Some of the suggestions in the comments are reasonable, but I think that, unless I need to express this relationship many times (in which case I'd probably explicitly make up a term and give a definition), I should avoid using a term that is not really widespread. $\endgroup$ – jdehesa Sep 6 '17 at 11:43

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