Math
A term for a function that depends on its point of evaluation (or other fixed parameters) as well as its argument? E.g. $\Delta f_{x}(\Delta x)$
Step-by-step mathematics solution: A term for a function that depends on its point of evaluation (or other fixed parameters) as well as its argument? E.g.
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1. Restating the Question in Plain Language
The student is looking for a standard name for an object that looks like
[ \Delta f_{x}(\Delta x)=f(x+\Delta x)-f(x) ]
or, more generally, an object that
- takes a fixed “parameter’’ (here the point (x) at which we are evaluating the increment), and
- also takes a variable argument (here the displacement (\Delta x)).
In other words, we have a function that depends on a point of evaluation (or other fixed data) as well as on its ordinary argument.
The question: What is the usual mathematical terminology for such an object?
2. Step‑by‑Step Explanation
Step 1 – Identify the two roles of the symbols
| Symbol | Role in the expression | Usual name in mathematics |
|---|---|---|
| (x) | a fixed quantity that determines which member of a whole collection we are looking at | parameter or index |
| (\Delta x) | the actual argument that we vary when we evaluate the object | variable (the “input’’) |
Thus (\Delta f_{x}) is a family of functions indexed by the parameter (x).
Step 2 – Write the object as a family (or parameterized function)
Define
[ \boxed{\;\Delta f : \underbrace{\mathbb{R}}{\text{parameter }x}\times\underbrace{\mathbb{R}}{\text{argument }\Delta x}\longrightarrow\mathbb{R},\qquad \Delta f(x,\Delta x)=f(x+\Delta x)-f(x)\;} ]
If we fix the first coordinate (x) we obtain a single‑variable function
[ \Delta f_{x}(\Delta x):=\Delta f(x,\Delta x). ]
Hence the notation (\Delta f_{x}) is just a partial application of the two‑variable function (\Delta f).
Step 3 – Standard terminology
| Context | Standard term(s) |
|---|---|
| A function that depends on an extra, non‑varying quantity is called a parameterized function or family of functions. | |
| When the extra quantity is used as an index (often discrete but can be continuous) we speak of a parameterised family indexed by that quantity. | |
| In functional‑analysis language, the map (\;f\mapsto\Delta f_{x}) is an operator (it takes a function (f) and returns another function of (\Delta x)). The specific operator here is the difference operator (or increment operator) at the point (x). | |
| In computer‑science / category‑theoretic language, fixing the first argument of a two‑argument function is called currying or partial application. The resulting object (\Delta f_{x}) is a curried version of the original two‑argument function. |
Consequently, the most common and widely understood term for the object the student is describing is:
A parameterized (or indexed) family of functions.
In the concrete example, ({\Delta f_{x}}{x\in\mathbb{R}}) is a family of increment functions indexed by the point (x), and each (\Delta f{x}) is the result of applying the difference operator at (x) to (f).
Step 4 – Connecting with the ellipse example
The ellipse notation
[ \vec{\mathscr{E}}:=\text{ellipse}[e,\mathbf f_{1},\mathbf f_{2}] ]
similarly defines a family of curves (or a parameterized surface) depending on the parameters (e,\mathbf f_{1},\mathbf f_{2}).
If we write
[ \mathscr{E}(t;e,\mathbf f_{1},\mathbf f_{2})\in\mathbb{R}^{2}, ]
then for each fixed triple ((e,\mathbf f_{1},\mathbf f_{2})) the map (t\mapsto\mathscr{E}(t;e,\mathbf f_{1},\mathbf f_{2})) is a single‑parameter function (the usual parametrisation of an ellipse). The whole object is again a parameterized family of functions.
Step 5 – Summarising the answer
- The object (\Delta f_{x}(\Delta x)) is a family of functions indexed by the parameter (x).
- The map ((x,\Delta x)\mapsto f(x+\Delta x)-f(x)) is a two‑variable function; fixing the first variable yields a parameterized (or curried) single‑variable function.
- The general term used in mathematics is “parameterized function’’ or “family of functions’’; the specific operator that creates it is the difference (or increment) operator at a point.
3. Final Answer
The standard name for an object such as
[ \Delta f_{x}(\Delta x)=f(x+\Delta x)-f(x) ]
or, more generally, a function that depends on a fixed “point of evaluation’’ (or any other fixed data) in addition to its ordinary argument is
[ \boxed{\text{a parameterized (or indexed) family of functions}} ]
Equivalently, one can say that (\Delta f) is a two‑variable function and (\Delta f_{x}) is the partial application (curried version) of that function, i.e. the result of fixing the parameter (x). The operator that produces it is called the difference (increment) operator at (x).
4. Common Mistakes
| Mistake | Why it’s wrong | Correct view |
|---|---|---|
| Calling (\Delta f_{x}) “a function of two variables” | The notation (\Delta f_{x}(\Delta x)) already has the parameter (x) fixed; it only varies with (\Delta x). | View (\Delta f) as a two‑variable function; (\Delta f_{x}) is a single‑variable function obtained by fixing the first variable. |
| Treating the parameter (x) as an ordinary variable | Then you would be mixing the roles of “parameter’’ and “argument’’ and lose the idea that each choice of (x) gives a different function. | Recognise (x) as a parameter (or index) that selects a member of a family of functions. |
| Calling it a “function of three variables’’ in the ellipse example | The ellipse is defined by a set of parameters ((e,\mathbf f_{1},\mathbf f_{2})) that are not meant to be varied simultaneously with the curve parameter (t). | See the ellipse as a parameterized family of curves: for each fixed ((e,\mathbf f_{1},\mathbf f_{2})) you get a single‑parameter map (t\mapsto\mathscr{E}(t;e,\mathbf f_{1},\mathbf f_{2})). |
| Confusing “operator’’ with “function’’ | An operator maps functions to functions; a function maps numbers (or vectors) to numbers. | (\Delta) is an operator (the difference operator). When applied to a specific (f) and a fixed (x) it yields a function (\Delta f_{x}). |
| Omitting the term “parameterized’’ | Without it the description is vague and does not convey that one variable is held fixed while the other varies. | Explicitly say “parameterized family of functions’’ or “family indexed by (x)’’ to capture the intended meaning. |
By keeping the distinction between parameters (fixed data) and variables (arguments) clear, one avoids these pitfalls.
Original question: A term for a function that depends on its point of evaluation (or other fixed parameters) as well as its argument? E.g. $\Delta f_{x}(\Delta x)$ on Mathematics Stack Exchange, licensed CC BY-SA.