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I am trying to express the following idea: "studying the variation of function f1 and the variation of function f2 w.r.t. the number of active users N is essential for ....". Note that we can use these functions instead of f2 and f1. Here is my attempt:

Studying the variation of these rate functions w.r.t. the number of active users N is essential for ...

In this attempt, should I use variation or variations? is or are ?

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  • Variation can be either a process or a result. So, the answer depends on what you study, the process of making changes to the function, or the multiple results of applying those changes.
    – Victor Bazarov
    Commented Nov 9, 2015 at 15:38
  • @VictorBazarov I am not sure I understand your point. Could you please elaborate on that a little further ?
    – tam
    Commented Nov 9, 2015 at 15:48
  • Example: "the function is subject to variation due to..." -- in this case it's a process, and is singular. Or, "in our experimentation with initial conditions we found at least three variations of the function..." -- here it's the result, and is plural.
    – Victor Bazarov
    Commented Nov 9, 2015 at 15:57

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This depends on how many differences are there between the two functions.

If there is a single difference use the singular 'variation' and 'is' otherwise use 'variations' and 'are'

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