# Questions tagged [functions]

For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.

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### Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
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### How to define a bijection between $(0,1)$ and $(0,1]$?

How to define a bijection between $(0,1)$ and $(0,1]$? Or any other open and closed intervals? If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that'...
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### What functions can be made continuous by "mixing up their domain"?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ such that $f\circ \phi$ is continuous. So one could say a potentially ...
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### Why do engineers use derivatives in discontinuous functions? Is it correct?

I am a Software Engineering student and this year I learned about how CPUs work, it turns out that electronic engineers and I also see it a lot in my field, we do use derivatives with discontinuous ...
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### How to straighten a parabola?

Consider the function $f(x)=a_0x^2$ for some $a_0\in \mathbb{R}^+$. Take $x_0\in\mathbb{R}^+$ so that the arc length $L$ between $(0,0)$ and $(x_0,f(x_0))$ is fixed. Given a different arbitrary $a_1$, ...
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### Nice expression for minimum of three variables?

As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function. $\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$ There's even a nice intuitive ...
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### Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$ [closed]

I'm thinking about this question in the sense that we often have a term $(-1)^n$ for an integer $n$, so that we get a sequence $1,-1,1,-1...$ but I'm trying to find an expression that only gives every ...
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### Create unique number from 2 numbers

is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must ...
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### What is the difference between $\arg\max$ and $\max$?

What is the exact difference between $\arg\max$ and $\max$ of a function? Is it right to say the following? $\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is ...
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### What is an operator in mathematics?

Could someone please explain the mathematical difference between an operator (not in the programming sense) and a function? Is an operator a function?
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### On the functional square root of $x^2+1$

There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$ If such $\phi$ exists (it does in this example), $\phi$ can ...
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### Is composition of measurable functions measurable?

We know that if $f: E \to \mathbb{R}$ is a Lebesgue-measurable function and $g: \mathbb{R} \to \mathbb{R}$ is a continuous function, then $g \circ f$ is Lebesgue-measurable. Can one replace the ...
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### What numbers can be created by $1-x^2$ and $\frac{x}{2}$?

Suppose I have two functions $$f(x)=1-x^2$$ $$g(x)=\frac{x}{2}$$ and the number $1$. If I am allowed to compose these functions as many times as I like and in any order, what numbers can I get to if I ...
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### Why is there no function with a nonempty domain and an empty range?

Let $A$ to be a nonempty set and $B= \emptyset$; then $A \times B$ is a set. And let $F$ be a function $A \to B$. Then $F \subseteq A \times B$. By the axiom of specification, $F$ must exists (if I ...
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### What is a basis for the vector space of continuous functions?

A natural vector space is the set of continuous functions on $\mathbb{R}$. Is there a nice basis for this vector space? Or is this one of those situations where we're guaranteed a basis by invoking ...
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### A function in which addition and multiplication behave the same way

Exponents have a well-known property: $$x^ax^b = x^{a+b}$$ but $$x^{a} + x^{b} \neq x^{a+b}$$ Similarly, $$\log(a) + \log(b) = \log(ab)$$ But $$\log(a)\log(b) \neq \log(ab)$$ So my question ...