1
vote
1answer
57 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
-1
votes
0answers
78 views

What is the name of formula?

Can someone help me to name this formula? $$ f(x) = \begin{cases} 1 + x & x \ge 0 \\ \frac{1}{1-x} & x < 0 \end{cases} $$ thanks.
1
vote
0answers
33 views

Different names for “function”

Quoting a book, "functions can also be named: Mappings, Transformations, Operators, Arrows or Morphisms" I have the idea that these different names are used depending on different contexts. But I ...
2
votes
3answers
137 views

Are the pre-image and the domain the same, or not?

Throughout school I thought that the pre-image was a subset of the domain, not that they were necessarily the same. When I spoke of a function f:R->R, I didn't think that this meant that f was defined ...
7
votes
6answers
1k views

What do I not understand about one-to-one functions?

Firstly, a definition: Definition 1: A function $\phi : X \rightarrow Y$ is one-to-one if $\phi(x_1) = \phi(x_2)$ only when $x_1 = x_2$. Now the question: Students often misunderstand the ...
2
votes
3answers
242 views

Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
1
vote
0answers
35 views

Addition, multiplication, exponentiation… What is next function of this series?

Addition can be (informally) defined as the application of successor function $S$ on $a$ $b$ times, i.e. $a+b=S\stackrel{b}{\cdots}S a$. Multiplication can be defined as the addition of $a$ with ...
0
votes
2answers
27 views

How do we emphasize that $\displaystyle x\mapsto\frac{1}{f(x)-y}$ “makes sense” if we know $y\notin\text{im }f$?

Please take a look at the following function $$x\mapsto\frac{1}{f(x)-y}$$ where $f$ is "some other function". Suppose we know $y\notin\text{im }f$, i.e. the expression in the denominator "makes ...
1
vote
0answers
14 views

Nominalization for being “not convex” and “not coercive”

Having a function f which is not convex or not coercive (coercive = |x| goes to infinity implies ...
0
votes
1answer
17 views

What's the difference between a partial function and a relation?

My understanding of a partial function is that it is one which only maps a subset of some set $A$ to another set $B$ (where $B$ could be $A$). On the Wikipedia page, the below image is given as an ...
1
vote
2answers
25 views

Name for a set of pairs of elements that equalise two functions?

Is there an established name for this $eql$ function? $$\operatorname{eql}(f, g) = \{\ (x, y)\mid f(x) = g(y)\ \}$$
0
votes
0answers
15 views

How to write the condition for Image of a function?

If $\Omega_l$ is $\Omega$ with $|x|<l$ and if $\Omega_S$ is the image of $z$ under mapping how we will write the condition for it. Am I right if I write $\Omega_S$ is $\Omega$ with $|S|<l$ or ...
0
votes
0answers
41 views

Why do we use the terms “non-increasing/non-decreasing/non-negative”?

I am not sure if I have to ask my question here. But I will try and thank you in advance. Why some authors (in books or in papers) use the following terms: Function $f$ is non-increasing; Function ...
0
votes
0answers
35 views

Are there official names for these functions?

$\newcommand{\sgn}{\operatorname{sgn}}$ Does anyone know if the simple function $$ y(x)=x^2\sgn(x)$$ or alternately $$ y(x)=x|x|$$ has any (official) name in mathematics or engineering? or ...
1
vote
2answers
24 views

Quick Question on Pre-image Terminology

Sorry for the daft question, but, is the following a correct thing to say? "The preimage of a function f is a function iff for any element b in the range, there exists exactly one a in the domain ...
0
votes
0answers
44 views

what is a degenerate function?

Consider the functin $f(x, y)$, e.g: \begin{align*} f(x, y) &= (x+y)^2 \\ f(x, y) &= (x+y^2)^2 \\ f(x, y) &= (35 \sin x+y^2)^2 \\ \end{align*} Dennis Auroux called this kind of ...
0
votes
0answers
80 views

About Kernel and the coimage of a function

Introduction I was serching for a concept of "equivalence relations" induced by an arbitrary function in a "natural" way and I found the concept of Kernel. But I'm not sure that I understand it and ...
0
votes
1answer
87 views

What are functions with the property $f(f(x)) = x$ called?

Do functions which, when composed with themselves, are equivalent to the identity function (i.e. functions for which $f(f(x)) = x$ in general) have a name and if so, what is it? Additionally, am I ...
0
votes
2answers
43 views

What is called the property of function that it does not change value when you transform arguments.

My question is probably rather simple, but I cannot find appropriate name or am too stupid to find a definition of such property. I'd like to give an example: $f({\bf r}_1, ..., {\bf r}_N, ) = ...
3
votes
0answers
30 views

Terminology Regarding Basic Properties of Functions

Is there a cultural difference between saying that a function is 1-to-1 or injective, onto or surjective and a 1-to-1 correspondence or bijective?
2
votes
1answer
676 views

into function vs injective function

In many mathematical books that I have read and from lectures from professors, the words 'into' and 'injective' were used interchangeably, but in Patrick Suppes book Axiomatic Set Theory he gives a ...
2
votes
2answers
109 views

What do you call a function with the property $f(-x)=-f(x)$?

What is this property called? The domain and codomain of the function can be for example $\mathbb Z^n$, $\mathbb Q^n$ or $\mathbb R^n$ ($n>0$), potentially excluding the $0$ point. Examples: ...
0
votes
1answer
46 views

Adding together curves or shapes to approximate something more complex

I'm looking for proper terminology / references for the following sort of problem: Say we have some one-dimensional curve like $y = 10$ defined over the real valued domain $[0,1]$, and we ask, how ...
1
vote
0answers
25 views

Terminology for functions with $F(a,a,\dots,a) = a$

Is there a commonly used way to call functions $F : \mathbb{R}^n \rightarrow \mathbb{R}$ such that if $x \in \mathbb{R}^n$ and $x_i = a$ for all $i\in \{1,\dots, n\}$, $F(x) = a$ ?
1
vote
2answers
86 views

Notation and terminology for functions, interpreting $f(y)$

It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean: Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also ...
1
vote
0answers
22 views

What can I use as the generic term for “a function that is composed with another”?

Suppose I am talking about the composition $g \circ f$ (or more generally $f_n \circ \cdots \circ f_1$). Is there a generic term for the functions $f$ and $g$ (the functions $f_i$)? "Compositand"?
0
votes
0answers
20 views

Definition of Range as Minimal Interval Containing Codomain

I am studying continuous functions where the domain is some interval (which may or may not be bounded, closed, etc). I am thinking about how continuity is related to other function properties, ...
1
vote
2answers
47 views

Matrix with Functions as Entries

What do we call a matrix with functions as entries? $$\textbf{f(x)}=\begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix} $$
0
votes
1answer
57 views

What is the terminology for this type of order-preserving function?

As I roughly know, a function $f$ in $\mathbb{R}$ is called order-preserving if $f(x)>f(y)$ for $x>y$, $x,y \in \mathbb{R}$. May I ask, if anybody knows the terminology for two functions $f$ ...
0
votes
0answers
26 views

Is there terminology to describe this type of function equivalence?

I have a binary vector, for simplicity of explanation, it contains +1's and -1's. Next I have a starting value, for example 5. As I iterate over the vector, my function takes my current value and adds ...
2
votes
0answers
43 views

Inverses of two argument functions with respect to one argument

Consider a function $f : A \times B \to C$ and two inverses, each with respect to one argument; i.e. $g$ and $h$ defined such that $f(x,y)=z \iff g(y,z)=x \iff h(z,x)=y$. A simple example is addition: ...
1
vote
2answers
105 views

Partial functions - where can I learn more about this (heuristic, informal) system of conventions?

Is there a name for the following (heuristic, informal) system of conventions for dealing with partial functions and undefined expressions? I'd like to know whether it has any undesirable quirks that ...
1
vote
1answer
52 views

Whats the name of this function?

I read this function in an exercise. It looks quit familiar to me, however I do not know its name. Whats the name of the $\rho_n$ function and who brought it up first?
6
votes
4answers
123 views

The limit of $f$ or the limit of $f(x)$?

I have read before that $f$ denotes the function $f$ whilst $f(x)$ denotes the value of the function $f$ at $x$. What is right? To say that the limit of $f$ as $x$ tends to $a$ is $L$ or to say that ...
1
vote
4answers
73 views

What is an odd function?

I'm reading this term (odd function) in my numerical analysis book, but I have never heard of this. What does it mean that an function is odd ?
1
vote
2answers
146 views

The bijective property on relations vs. on functions

I recently encountered what seems to me like an inconsistency in the usage of the term bijective. This is pretty basic stuff, but it somehow never occurred to me before. I'd like to make sure I'm ...
3
votes
1answer
89 views

One-to-one mapping vs one-to-one correspondence

Does the phrase "one-to-one mapping" mean the same thing as "one-to-one correspondence?" I know that the latter refers to a bijection. Does the former refer to an injection (i.e. it is the same as ...
1
vote
0answers
38 views

Condition for being 'injective in each variable separately'

Is there an established term for a function on a product set that is not injective but is injective with respect to each argument individually? The motivating example is the canonical projection $\pi: ...
2
votes
1answer
81 views

Terminology for $\phi(xy)=\phi(x)\phi(y)$

I have a model which contains a function $\phi:{\mathbb R}_+ \rightarrow {\mathbb R}_+$ that satifies: $$\tag{*}\phi(xy)=\phi(x)\phi(y)$$ for all $x,y\in{\mathbb R}_+$. In Number Theory there is a ...
6
votes
3answers
173 views

Unambiguous terminology for domains, ranges, sources and targets.

Given a correspondence $f : X \rightarrow Y$ (which may or may not be a function) I generally use the following terminology. $X$ is the source of $f$ $Y$ is the target $\{x \in X \mid \exists y \in ...
19
votes
4answers
399 views

“$f$ is a function from $A$ to $B$” vs. “$f $is a function from $A$ into $B$”?

When we say that $f$ is a function from $A$ to $B$ is this different from saying $f$ is a function from $A$ into $B$ I know what injective ("1-1"), surjective ("onto"), and bijective ...
2
votes
1answer
72 views

Is there a name for the function $(1 - e^{ct})/(1 - e^{c})$?

$$f(t) = \frac{1 - e^{ct}}{1 - e^{c}}$$ This is a function which is somehow a streched exponential which is zero at $t = 0$, and one at $t = 1$, where $c$ determines the curvature (with $c = 0$, it ...
5
votes
1answer
101 views

Is the variant direct image mathematically significant?

Preimages have the property that for an arbitrary function $f : X \rightarrow Y$ and all $B \subseteq Y$ it holds that $$f^{-1}(B^c)=[f^{-1}(B)]^c.$$ However, the analogous statement for direct ...
3
votes
0answers
55 views

What's the mathematical field called where functions create and delete functions?

Motivation In the field of modular, reconfigurable robotics there are some groups which use term rewriting, or specifically graph rewriting to describe the reconfiguration process of the modular ...
1
vote
5answers
704 views

How do I find the image of the functions $y=2$ and $y = 2x - 6$?

The function is $y=2$, the domain is just 2? And the image of it? I don't think I quiet understand what the image of a function means, the domain is all values that it can assume, correct? Could you ...
1
vote
1answer
1k views

Definition of correspondence

A one-to-one correspondence is an alternative name for a bijection between two sets, but to what does the term 'correspondence' alone refer? As far as I can see, it seems to be another term for ...
1
vote
1answer
50 views

Limit of a seqence $\{f_n \}_{n\in \mathbb N}$ of functions?

I don't really know how mathematicians talk about this concept. I try to explain better what I mean with limit of a sequence of functions: Given a countable set of functions $\{f_n \}_{n\in \mathbb ...
1
vote
2answers
65 views

correct name of mathematical property

I am developing a program that transforms artifacts in one (computer) language to artifacts in another language. In my program there are certain border line situations where the result of applyin the ...
1
vote
1answer
42 views

How can I name a $(a+x)\cdot b$ function

I'm looking for a name to identify functions like $f(x) = (x + a)\cdot b$. Does this kind of function have a specific name (like "affine" for $a\cdot x+b$)?
2
votes
1answer
80 views

Is there a common name for the property $f(\lambda x+(1-\lambda)y)\leq f(x)+f(y)$?

For $X$ a vector space and $f:X\to\mathbb{R}$, I'm considering the property $$f(\lambda x+(1-\lambda)y)\leq f(x)+f(y) \qquad\forall x,y\in X, \;\forall\lambda\in[0,1].$$ So it's like convexity ...