Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
34,642 questions
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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?
195
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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's ...
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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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Identification of a curious function
During computation of some Shapley values (details below), I encountered the following function:
$$
f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}},
$$
where $...
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Is there a bijective map from $(0,1)$ to $\mathbb{R}$? [closed]
I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
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8
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How do I prove that a function is well defined?
How do you in general prove that a function is well-defined?
$$f:X\to Y:x\mapsto f(x)$$
I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
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votes
7
answers
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Functions that are their own inverse.
What are the functions that are their own inverse?
(thus functions where $ f(f(x)) = x $ for a large domain)
I always thought there were only 4:
$f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ ...
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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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6
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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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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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Is there a way to get trig functions without a calculator?
In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a ...
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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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Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
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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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