Elementary questions about functions, notation, properties, and operations such as function composition.

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0
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3answers
99 views

Example of a function $f: [a, b] \to \mathbb{R}$ that is unbounded.

Can anyone give me an example of a function $f\colon[a,b]\to\mathbb{R}$ that is unbounded?
1
vote
2answers
363 views

Sequence of Uniformly Bounded functions

Consider a sequence $\{ f_k \}_{k=1}^{\infty}$ of locally-bounded functions $f_k: \mathbb{R}^n \rightarrow \mathbb{R}_{\geq 0}$. Assume the following. For any sequence $\{X_k\}_{k=1}^{\infty}$ of ...
0
votes
2answers
296 views

General method to find inf, sup, maxs and mins of a function

Could someone explain how to find inf, sup, max and min values of a function (real-valued functions of real variable, generally continuous/differentiable, with some possible points of discontinuity)? ...
2
votes
2answers
92 views

Studying this function: $y = \frac{x^2}{1+\log|x|}$

I'm still studying this function: $$y = \frac{x^2}{1+\log|x|}$$ And now I'm dealing with the study of the monotony. So I got the first derivative, this: $$y\,' = \frac{x(1+\log{x^2})}{(1+\log|x|)^2}$$ ...
1
vote
0answers
112 views

Functions realized by perceptrons.

Consider all functions $f : \{-1, 0, 1\}^3 \to \{-1, 1\}$. How many of these functions exist and how many can be realized by a perceptron. What are the conditions that one has to check? N.B. A ...
1
vote
5answers
235 views

Any fastest algorithm for $f(n) = f(n-1) \cdot f(n-2)$ where $3 \leq n \leq 1000000$

Any fastest algorithm for $$ f(n) = f(n-1)\cdot f(n-2)\quad\text{ where }\quad f(1) = 1,\quad f(2) = 2 $$ for $3 \leq n \leq 1000000 $.
9
votes
5answers
1k views

Is it possible to combine two integers in such a way that you can always take them apart later?

Given two integers $n$ and $m$ (assuming for both $0 < n < 1000000$) is there some function $f$ so that if I know $f(n, m) = x$ I can determine $n$ and $m$, given $x$? Order is important, so ...
1
vote
1answer
93 views

Is the inverse of a one-to-one, total function itself one-to-one and total?

The problem, from Boolos and Jeffrey's Computability and Logic, states the following definitions: "If a function $f(a)$ (from $A$ to $B$) is defined for every element $a$ of $A$, then it is called ...
3
votes
4answers
14k views

How to prove if a function is bijective?

I am having problems being able to formally demonstrate when a function is bijective (and therefore, surjective and injective). Here's an example: How do I prove that $g(x)$ is bijective? ...
0
votes
3answers
231 views

What is the inverse of $f(n)=\frac{n^2+n}{2}$?

I'm building an algorithm to determine whether a value is inside a series. To speed it up, I need the inverse function of the following series: $$1 + 2 + 3+\cdots +n$$ What is the inverse function ...
0
votes
1answer
51 views

Algorithms and generalisation of functions

I admit I'm little bit poor in functions in mathematics. But I'm in real urge to get this riddle out. How to express $$x(n)=x(n-1)+x(n-2)+1,$$ where $n>1$ and $x(0)=0$ and $x(1)=1$, in terms of ...
4
votes
1answer
150 views

Algebra of Functions

There seems to be an interesting algebra of functions. Does it already exist in literature? Given functions $f_1 : X_1 \to Y$ and $f_2 : X_2 \to Y$, if $f_1(x) = f_2(x)$ for all $x \in X_1 \cap X_2$, ...
0
votes
1answer
241 views

Produce function of curve, given some coordinates

I tried asking something similar to this before, but I guess I didn't explain it very well. Hopefully this will be more clear. I need to get an equation which describes a curve with certain points. ...
6
votes
2answers
113 views

Representation of smooth function

Is it true that any smooth function $f\colon \mathbb{R}^n \to \mathbb{R}^n$ can be represented as $$ f(x) = \nabla U(x) + g(x) $$ where $U(x)$ is a scalar function and $\langle g(x), f(x) \rangle ...
2
votes
2answers
265 views

Comparing and contrasting equations and functions

I have several related questions, so I'm going to label them to make sure I understand what questions that answers are referring to. I understand that a function is an expression that produces one ...
2
votes
1answer
399 views

'Plus' Operator analog of the factorial function? [duplicate]

Possible Duplicate: What is the term for a factorial type operation, but with summation instead of products? Is there a similar function for the addition operator as there is the factorial ...
4
votes
3answers
599 views

If $f \circ g$ is invertible, is $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$?

If $f \circ g$ is invertible, is $(f \circ g)^{-1} = g^{-1} \circ f^{-1}$? If not can someone give me a counterexample?
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2answers
211 views

Piece-Wise Discontinuity & Continuity

When is the following function continuous? How would i go about listing the removable discontinuities and then redefine the function so that it is now continuous in those places? $$f(x)= \begin ...
3
votes
0answers
180 views

Prove an image of function $f:[a,b]\to\mathbb R^n:t\mapsto(f^1(t),f^2(t),\ldots,f^n(t))$ where $f^i\in C^\infty$ doesn't contain open ball

$\mathbb R^n\supset[a,b]$ is domain of definition $f:[a,b]\to\mathbb R^n:t\mapsto(f^1(t),f^2(t),\ldots,f^n(t))$ where $f^i\in C^\infty$ I need to prove that image of $f$, that means $f[a,b]$, doesn't ...
1
vote
1answer
158 views

What kind of analytical function could resemble that plot?

I am in the middle of making a model, and I am looking for an analytical expression which could resemble this evolution for one of the parameters:                 In short: a sudden increase from 0 ...
4
votes
1answer
189 views

Simple functional equation. Find $f:\mathbb Q\longrightarrow\mathbb Q$ (own)

Find the functions $f : \mathbb{Q} \mapsto \mathbb{Q}$ knowing that $$2f\left(f\left(x\right)+f\left(y\right)\right)=f\left(f\left(x+y\right)\right)+x+y,\ \forall x,\ y\in\mathbb{Q} $$
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votes
1answer
1k views

how to rotate a Gaussian?

Lets suppose that we have a 2D Gaussian with zero mean and one covariance and the equation looks as follows $$f(x,y) = e^{-(x^2+y^2)}$$ If we want to rotate in by an angle $\theta$, does it mean ...
4
votes
2answers
141 views

cardinality of $E^F$, $E, F = \emptyset$

My textbook (Naive Set Theory) asks the reader to show that $\left| E^F \right|$ = $\left| E \right| ^ \left| F \right|$ for all finite sets. In passing to induction, I noticed that this would imply ...
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vote
2answers
96 views

$p(x) \mid q(x)$ for infinite values of $x$ (integer) implies $p(n) \mid q(n) \quad \forall n$ integer

I was working on: Kind of Functional Eq. in Integers I found a sort of way... but I need to show that: $p(x) \mid q(x)$ for infinite values of $x$ (integer) implies $p(n) \mid q(n) \quad \forall n ...
2
votes
1answer
55 views

Vector as argument of a function

Given a function $f(x)=y$ is correct to say that $f\left(\left[\begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array}\right]\right)=\left[\begin{array}{c} y_1 \\ y_2 \\ y_3 \end{array}\right]$?
1
vote
1answer
181 views

Maximum point of a polar function

I have a curve C with polar equation $$r^2 = a^2\cos{2\theta} $$ And I am looking to find the length $x$ when $r=max$ Judging from the equation: $$r = \sqrt{a^2\cos{2\theta}} $$ R will be maximum ...
2
votes
0answers
125 views

How can we glue Sobolev functions?

Let $u:A\cup B\to R$ be a function (where $A$ and $B$ are disjoint connected sets and $A\cup B$ is connected) such that $u$ restrict to $A$ and to $B$ are in $W^{1, p}$. Which result guarantees me ...
1
vote
1answer
162 views

generalised inverse function

Let $f:\mathbb{R} \rightarrow [0, 1]$ be increasing (edit: i.e., non-decreasing). Define $f^-(y) = \inf \{x \in \mathbb{R} : f(x) \geq y \}$, $y \in [0, 1]$. Is the following line true? $$x \leq ...
35
votes
5answers
4k views

How do I define a bijection between $(0,1)$ and $(0,1]$?

How do I 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 ...
1
vote
0answers
32 views

Ask a question about quasisymmetric maps

Is there a function which maps the Cantor ternary set onto itself , such that it is a quasisymmetric maps but not a Lipschitz maps ? If the answer is yes, give a example.
6
votes
10answers
409 views

Sanity check, is $\{(-9,-3),(2,-1),(7,7),(-1,-1)\}$ a function?

EDIT#2: Yes, I'm crazy! This IS a function. Thanks for beating the correct logic into me everyone! I'm using a website provided by my algebra textbook that has questions and answers. It has the ...
1
vote
0answers
70 views

How to find a function with the following properties?

I want to find a function $f(s,x)$ such that $f(s,x)$ is analytic for any $s \in Z^+ $, $f(s,x)=B_s(x)$, where $B_s(x)$ are the Bernoulli polynomials $f(a, x)$ is elementary against $x$ at any ...
13
votes
6answers
8k views

Can there be a function that's even and odd at the same time?

I woke up this morning and had this question in mind. Just curious if such function can exist.
4
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4answers
1k views

show if function is even or odd

Suppose that we have equation: $$f(x)=\frac{2^x+1}{2^x-1}$$ There is question if this function even or odd? I know definitions of even and odd functions, namely even is if $f(-x)=f(x)$ and odd is if ...
0
votes
1answer
293 views

Find the period of the following function

suppose that we have function $y=[2x]-3*[4x]$ here $[*]$ denotes as a minimum distance till integer. we are required to find period of this function,first of all i am confused in terms of ...
3
votes
1answer
113 views

Hypergeometric functions inequality

Let $_2F_1(a,b;c,z)$ be the (Gauss) hypergeometric function, and $m$ and $n$ positive integers. From a simple plot it looks like $_2F_1(m+n,1,m+1,\frac{m}{m+n})>\frac{m}{n} ...
1
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1answer
86 views

Invert “Gravitational” Force Function or Solve an Intersection

Recall "gravitational"-type force functions, by which I mean anything of the form: $f(x,y,z) = \frac{k}{((x-x_0)^2+(y-y_0)^2+(z-z_0)^2)^p}, p\in\Re_{>0}, k\in\Re, (x,y,z) \neq(x_0,y_0,z_0)$ (e.g., ...
5
votes
2answers
75 views

Can such a function exist?

Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$. Does there exist a function $f: \mathbb{R} \to ...
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2answers
247 views

What is the underlying function?

I have some value pairs. They are inversely proportional. I want to have a formula to get the second value from the first value. ...
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2answers
154 views

Function that magnifies small changes and compresses large changes

I need a function (for a heatmap algorithm) that takes a percentage difference between two values, and returns a number between 0 and 1. The output will be used in coloring parts of the screen. The ...
9
votes
2answers
470 views

Finite groups of functions under function composition

Over the years I have done many questions along the lines of the following: "Given functions $\phi, \theta$ (usually defined on $\mathbb{R}$ or $\mathbb{C}$, or a suitable subset of $\mathbb{R}$ or ...
0
votes
1answer
72 views

Ordering some functions without losing continuity

Let's consider the set $\mathbb R^{n}/S_n$, i.e. the quotient of $\mathbb R^{n}$ modulo permutations. An element $a \in \mathbb R^{n}/S_n$ is simply a $n$-tuple of real numbers (the order does not ...
1
vote
1answer
91 views

How do I determine whether a function is operator monotone on a given interval?

Just to avoid confusion, a function is called matrix monotone in an interval $[a, b]$ if $A - B \geq 0$ implies $f(A) - f(B) \geq 0$ for any Hermitian Matrices $A, B$ (we can restrict to finite ...
6
votes
0answers
245 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
6
votes
1answer
314 views

Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.

The Olympiad-style question I was given was as follows: A function $f:\mathbb{N}\to\mathbb{N}$ is defined by $f(1)=1$ and for $n>1$, by: ...
2
votes
1answer
48 views

Terminology for a function computed by a finite-state transducer?

A finite-state transducer is a generalization of a finite state machine that accepts an input string and produces an output string (instead of just accepting or rejecting). Is there a name for a ...
1
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0answers
67 views

Is there chance to form a frame (Riesz basis)?

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ One can show that ...
0
votes
1answer
197 views

How to “stretch” a procedural half-sphere texture on X and/or Y axis

I've implemented an Objective-C function to display the "height" of a half-sphere, with "1.0" being "full-height" and "0.0" being "no-height" The sphere currently has a few parameters: Center (x,y: ...
2
votes
1answer
213 views

When are arbitrary constants defining function families independent?

I'm not sure what the proper terms are here, so I figure it's better to illustrate with examples. If I look at the family of polynomials of a certain degree (e.g cubics), the coefficients in front of ...
14
votes
2answers
372 views

Increasing orthogonal functions

What is the maximal $n$ such that there exist functions $f_1, \dots, f_n:[0,1] \to \mathbb{R}$ that are all bounded, non-decreasing, and mutually orthogonal in $L^2([0,1])$?