# Tagged Questions

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### 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: ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### Translating text to functions

I am having problems understanding how to extract this information into a formula. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Name for a set of pairs of elements that equalise two functions?

Is there an established name for this $eql$ function? $$\operatorname{eql}(f, g) = \{\ (x, y)\mid f(x) = g(y)\ \}$$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...