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“A variable so related to another that for each value assumed by one there is a value determined for the other.” (thefreedictionary.com)

I can’t find any possible way how to start with this words. I’d like to say what I don’t know at least, but only to fail to do so. I hope to be got the right parse and its meaning.

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    The meaning of function in this case should has the same meaning as one in this example: The degree of drought is largely a function of temperature and drainage. And variable in the original sentence means a quantity that may assume any one of a set of values. The original sentence's construction should be "a variable so ... that ...".
    – Searene
    Jun 5 '14 at 4:40
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First of all, note that, like most dictionary definitions, this is not a complete sentence. To make it a complete sentence, we can read it as "[A function is] a variable so related...".

Second, note that the noun "variable" is implied in the prepositional phrase beginning with to, and in a few other places thereafter:

A variable so related to another [variable]...

Third, I think you may be getting tripped up by the adverb so, which is used in a nonintuitive way in this definition. It is used here in the sense of "in such a way" or "in the manner indicated":

A variable [that is] related to another [in such a way] that for each value assumed by one there is a value determined for the other.

This is roughly the same sense in which you use the word yourself: "...only to fail to do so."

The reason it's confusing here is that although the word "related" is being used here as a verb in the past tense, "related" can be used as an adjective as well. And when we see the construction "[noun] ([verb]) so [adjective]," the word so is usually used in a different sense--that of an intensifier indicating a greatness of degree:

...a man so old his hair was as white as snow.

Knowing all this, we can rewrite the definition in a way that may be clearer to you:

[A function is] a variable [that is] related to another [variable] [in such a way] that for each value assumed by one [variable] there is a value determined for the other [variable].

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