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I would like to know what good and valid ways there are to say (in words) that some value f(x), which depends on a variable x, in fact only depends on x "through" some function of x.

Example: For $x\in\mathbb{R}$ let $\hat{x}:=\min(0,x)$ and let f be a function such that $f(x)=f(\hat{x})$ for all x. I want to express that "f only depends on x 'through' $\hat{x}$".

How to formulate this (in words rather than writing it down in formulas)? While for the above example it may be easier to just write down the formula, there can of course be more complex situations, in particular when not dealing with numbers, for example that the expectation of a random variable "depends on the r.v. only through its distribution".

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You could say that $f(x)$ is "determined" by $\hat x$. That might not be clear to everyone, so you should spell out that $f(x_1)=f(x_2)$ whenever $\hat x_1=\hat x_2$. For example, if two random variables have the same distribution, then they have the same expectation. – Rahul Mar 11 '13 at 12:14
With morphisms instead of functions, one would say that "$f$ factors over $x\mapsto \hat x$" – Hagen von Eitzen Mar 11 '13 at 12:51

Translating the formulas to words is one way: f is a function of x such that if we denote g(x) by y, f can be written as a function of y only. Perhaps not exactly what you are looking for.

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