Right preposition for: Evaluation of a function at 0 “in/at” its first argument [closed]

Consider a function f(x,y) defined on X x Y.

One says that f(0,y) is the evaluation of f at 0 ...

1. ... in its first argument
2. ...at its first argument
3. (none of these)

My problem using searching engines is that usual occurrence of evaluation is evaluation at certain point, but if the function in question has multiple arguments, then which preposition helps to specify the argument it is evaluated in/at?

closed as off-topic by BillJ, Nathan Tuggy, FumbleFingers, Eddie Kal, David RicherbyNov 19 '18 at 17:56

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• A homework question? – BillJ Nov 16 '18 at 16:30
• @BillJ sounds more like c.p. is the teacher trying to write the question. – Andrew Nov 16 '18 at 16:32
• It's not clear how you can "evaluate" a function that has two arguments when you only provide the one argument -- or rather the result will be some range of values. So I'm not sure that "evaluate" is even the right term for this. It's been a long time since I took any serious math classes. – Andrew Nov 16 '18 at 16:34
• I'm voting to close this question as off-topic because it's about specialized terminology that most native speakers would need to look up. – Nathan Tuggy Nov 16 '18 at 16:58
• You probably want the value of f with zero as its first argument. The preposition "at" leads us to believe that there's a single argument -- even if that one argument is a set of values (e.g. coordinates). That's why you're getting so many search engine results for "evaluation at a certain point". – Gary Botnovcan Nov 16 '18 at 17:26

I am not sure that there is a unique way to translate f(0, y) into English. The whole point of mathematical notation is to provide a succinct and less ambiguous form of communication than any natural language. Moreover, there are usually many ways to convey the same thought in a natural language.

In most cases, I would avoid the word "evaluation" because it implies a numeric answer. If there is no numeric answer, "evaluation" creates a misleading expectation. "f(0, y) is the simplification of f(x, y) in the special case that x equals zero" seems to me to work, but there are probably hundreds of ways to express the same thought in English.

Math has its own unique way of saying things, so this isn't really a general-purpose English question. I'm not sure if "evaluate" is the right verb to use when talking about defining a function of two variables when only one variable is known, as the result is another function, not a specific value.

Which is why I have some trouble figuring out what to say, since it's not clear what you're trying to do. For example if I were to try to explain the problem I might say something like:

Given a function f(x,y), then f(0,y) is a reduction of the function when x=0. Similarly we can say f(n,y) reduces/simplifies the function when the first argument is known to be n.

My verbiage is probably a little strange, as it's been a long time since I've taken any serious math courses. Someone who does this on a regular basis could better tell you what terms would be common.

I assume from the context that this question concerns a function from two-dimensional real space (RxR) into the real line R. That assumption helps me explain my answer but does not change it.

According to Burkill and Burkill A Second Course in Mathematical Analysis, the word that describes a function, such as f(0, y), which takes values that are identical to another function (in this case f(x, y)) but on a subset of the latter's domain (in this case the line x=0 is a proper subset of RxR) is the restriction of f to the the subset.

I daresay there are other terms used by other authors. In quite a few years of studying mathematics I myself have not encountered in this context either 'simplification' (pace @JeffMorrow) or 'reduction' (pace @Andrew) but it is quite possible that they are used also.

If you want to refer specifically to the values taken by f(0, y), they are its range.

• That's right. However, if you are interested in the algebra A of all real valued functions defined on X x Y (yes,suppose X, and Y are subsets of the real line R), instead of a fixed function, and you consider the map Ev_{p,q}: A----> R, given by Ev_{p,q}(f) = f(p,q), for fixed points p,q of X and Y respectively, traditionally in functional analysis this is called evaluation. My doubt is not about this terminology (I am 100% sure of) but about the preposition. – c.p. Nov 18 '18 at 8:53
• "for fixed points p,q of X and Y". You did not say in your question that y was a fixed point. If it is, then I am in no doubt that the preposition is "at". And even if y is not a fixed point in R, (0, y) is a point in the set of lines in the space XxY, so you could still use "at (0,y)". I do have doubts about using the words "...first argument". I do not recall ever seeing that usage. – JeremyC Nov 18 '18 at 13:35