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

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7 views

How do we call a pair of sets between which there is a bijection that need not have additional property?

Let $A,B$ be sets. It is conventional to say that $A,B$ are isomorphic if there is a linear bijection between $A$ and $B$; that $A,B$ are homeomorphic if there is a continuous bijection between $A$ ...
0
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2answers
36 views

Intuitive explanation of the Dirichlet function and rationality

The Dirichlet function is defined by $f(x)=\begin{cases} c &\text{ if } x\in \mathbb{Q}\\d &\text{ if } x\notin \mathbb{Q}.\end{cases}, c\neq d$ See MathWorld's page for the full definition. ...
3
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0answers
9 views

Characterize in terms of fibre

I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you. For a function $f:X\to Y$, and y an element of ...
2
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2answers
35 views

What function is this? $\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$ [on hold]

I am interested to know what function represents the following series: $$\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$$
4
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1answer
38 views

A continuous bounded function from $\mathbb R$ to $\mathbb R$ can be increasing or not?

Let $f:\mathbb R \rightarrow \mathbb R$ be a continuous and bounded function , then $a$) $f$ has a fixed point. $b$) $f$ cannot be increasing $c$) $\lim_{x\rightarrow \infty} f(x)$ exists. ...
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1answer
19 views

General mathematical induction statement for establishing any result. [on hold]

I do not know even how to correctly frame this question. Normally a specific result is given and it is required to prove this specific result by induction. But How does one symbolically generalize ...
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2answers
51 views

Periodic function without trigonometry and complex numbers [on hold]

Can I get a periodic function without using trigonometric functions or complex numbers?
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2answers
63 views

Are all operations functions?

I have looked at Wikipedia(I know it's not completely reliable) but on it an operation is formally defined as: "A function ω is a function of the form $ω : V → Y$, where $V ⊂ X_1 × … × X_k$." and I ...
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2answers
23 views

Fibers of an Element

Prepping myself for a graduate abstract course and we are using Dummit and Foote's Text. We are starting on Chapter 1, so I thought it would be a good idea to go over chapter 0. There is a term that ...
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3answers
47 views

Injective function $g:B \to A$ from a surjective function $f:A \to B$

I wish to prove the existence of an injective function $g:B\to A$ given a surjective function $f:A\to B$. This sounds simple enough, however I'm having trouble writing a formal proof for it. Thanks ...
2
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0answers
23 views

decomposing a function into embedding and projection

I have a simple question. If $f:\mathbb{S}^{2}\rightarrow\mathbb{R}$ is a non-constant continuous function, can we represent it as a composition $f=p\varphi$, where ...
0
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1answer
17 views

Limit of a Monotonic Increasing and Non-Bounded Function

I have made a solution for the following question and I'm wondering if it's correct. I think that something is missing here. Can you help me complete the solution? Let $f$ be a function. The ...
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0answers
10 views

About the stable/invariant point sets in a plane with respect to shift/linear transformation

I'm reading Vlademir A. Zorich's Mathmatical Analysis I, meeting exercise question as following: a) A set $S \subset X$ is stable with respect to a mapping $f:X \rightarrow X$ if $f(S) \subset ...
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3answers
28 views

Equality of One-One functions. [on hold]

We're given two functions $f$ and $g$ , both $f$ and $g$ are one-one and onto. Now, if we say that $f(x) = g(x)$ for a value of $x$, can we conclude that $f=g$ ?
3
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2answers
24 views

Minimum of $f(x)=\sum_{i=1}^n\frac{a_n}{x-b_n}$ occurs at extreme point?

Let $a_1,\ldots,a_n$ be real numbers and $b_1,\ldots,b_n>1$. Define $$f(x)=\sum_{i=1}^n\frac{a_i}{x-b_i}.$$ Is it always true that $f(x)\geq\min\{f(0),f(1)\}$ for all $x\in[0,1]$?
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1answer
32 views

Does anyone know of an entire function $f$ such that $f(x,y)=0\iff\exists n\in\Bbb{N}\left[y=nx\right]$?

I tried constructing $f$ as $f(x,y) = n\left(\frac{y}{x} \right)$ where $n(a)=\lim \limits_{x \to a} \frac{\sin(\pi x)}{x}$, but I can't figure out how to remove the discontinuities when $x$ or $y$ is ...
0
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1answer
14 views

Classify the growth of functions and find a more general growth function

The following function $f(t,x):[0,T]\times R\mapsto R$ such that $\int^T_0|f(t,0)|^2 d t<\infty$, where $0<T<\infty$. If $f(t,x)$ satisfies $|f(t,x)|\leq Ax+B$ for each $x\in R$ and $A, B$ ...
0
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0answers
23 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
0
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1answer
46 views

How many distinct roots $ax^5+bx^3+cx+d$ has

$a,b,c>0$ How many distinct roots $ax^5+bx^3+cx+d=0$ has? question doesnt clarify which kind of root it has. and I dont understand why the question didnt say 'may has' . because by ...
0
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0answers
19 views

Can I get anywhere with working out an unknown function if input is restricted? [on hold]

I want to know the formula that a game is using for some task (doesn't really matter what exactly). I know what the inputs and outputs are, but there's a finite number/combination of inputs - I know ...
0
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0answers
20 views

How to obtain accumulated counts of past events by time $t$?

Given $f: [0, \infty) \to \{0,1\}$, $f(t)$ represents whether there is an event occurring at time $t$. How can we obtain $g: [0,\infty) \to \mathbb{N}_0$ so that $g(t)$ represents the number of ...
10
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4answers
206 views

Find all functions f such that $f(f(x))=f(x)+x$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(f(x))=f(x)+x, \forall x\in\mathbb{R}$. Find all such functions $f$. Clearly, $f$ is an "one-to-one function". I have tried setting ...
0
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1answer
15 views

Find maxima and minima of the function

Given: $$f:\mathbb{R}^2 \rightarrow \mathbb{R}, f\left(x,y \right)=-x^4+x^3-3x^2y+3xy^2-y^3$$ Find all points where gradient is equal to zero. Decide whether in those points function has either maxima ...
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2answers
24 views

Map real numbers into [0:255] using fixed limits interval

I got an interval from x0 to x1. I want all numbers inside this interval to be mapped (preferably linear) from 0 to 255. All numbers below x0 should be mapped to 0. All numbers above x1 should be ...
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5answers
55 views

Are these two expression equal?

My friend insisted that $(-1)^{(-n)}$ is equivalent to $(-1)^n$ for any number of $n$. A quick check in the Wolfram Alpha show ...
1
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2answers
47 views

finite vs infinite set function composition

If there is a set $X$ which is finite with $f : X \rightarrow X$ and $g: X \rightarrow X$, then $f \circ g = 1_X$ iff $g \circ f = 1_X$. How is it true for finite sets? I'm not too sure, but the ...
1
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1answer
31 views

Can true surjection really exist for algebraic functions?

Quoting a definition from Wikipedia: A surjective function is a function whose image is equal to its codomain. Consider an arbitrary algebraic function that has $\mathbb{R}$ as its codomain. ...
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0answers
9 views

Tweaking function to reduce the rate of decay of a logarithmic based curve

Im not even sure if this is possible or perhaps I may need to use a different function altogether but I currently have one that looks like this: $$y = a\log(x+b)+c$$ That produces the red curve ...
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0answers
10 views

trying to use function transformations with ln

I'm trying to understand if I can use function transformations in the usual sense (vertical and horizontal shift, stretching, etc...) with ln. Specifically, if I look at the graph of ln(x^2) and then ...
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4answers
50 views

Show that the function is continuous

To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}$ with $f=\left\{\begin{matrix} \frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\ 0 & , (x,y)=(0,0) \end{matrix}\right.$ is ...
0
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1answer
33 views

What's wrong with this proof for all subsets A and B of X, $F(A\cap B)=F(A)\cap F(B)$?

Definition: If $F:X \rightarrow Y$ and $A\subseteq X$, then $F(A)=\{y\in Y|y=F(x)\text{ for some x in A}\}$ Proposition For all subsets A and B of X, $F(A\cap B)=F(A) \cap F(B)$ Let $F$ ...
3
votes
3answers
157 views

How can the trigonometric equation be proven?

This question : Whats the size of the X angle? has the answer $10°$. This follows from the equation $$2\sin(80°)=\frac{\sin(60°)}{\sin(100°)}\times \frac{\sin(50°)}{\sin(20°)}$$ which is indeed ...
0
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2answers
24 views

How can I complete my solution in function problem?

Let $f:\mathbb R\to \mathbb R$ with $f(x)f(y) - f(xy) = x + y$ for every $x,y \in R$. Prove that: a)$f(0) = 1$ b)$f(x) = x + 1$ My solution: a) $f(x)f(y) - f(xy) = x + y$ ...
11
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2answers
77 views

Does there exist a function $g\in \mathbb{N}^\mathbb{N}$ s.t. $\{f\mid f\circ f=g\}$ is not empty and finite?

I'm struggling with this question and can't figure it out. The question was too long for the title so I will write it once more: Does there exist a function $g : \mathbb{N} \longrightarrow ...
0
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1answer
36 views

For two functions $f,g$ from $X$ to $\mathbb R$ how should I interpret $f \wedge g$?

For two functions $f,g$ from $X$ to $\mathbb R$ how should I interpret $f \wedge g$? I came to such a term while reading the axioms of fuzzy topology.
3
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2answers
43 views

Number of integer functions satisfying three constraints

I am trying to understand how many functions $\mathbb{Z^+}\to \mathbb{Z^+}$ which satistfy the three following constraints exist: For every $n \in \mathbb{Z^+}$ $$f(f(n))\leq\frac{n+f(n)}{2}$$ For ...
2
votes
4answers
61 views

Random number function (counting)

I have task I can't get my head around, even with a suggested answer. You have a function the generates a random integer between $0 - 65535$. Your task is to generate random integers $125-525$ ...
3
votes
2answers
30 views

Proving a function $F$ is surjective if and only if $f$ is injective

Problem: Let $X$ and $Y$ be non-empty sets and let $f: X \rightarrow Y$ be a function. Then we can define $F: P(Y) \rightarrow P(X)$ by \begin{align*} F(B) = f^{-1}(B) \qquad \text{for all} \ B \in ...
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1answer
70 views

define two functions whose compositions are equal to identity

Let B be the set $B = \{1,2,....n\}$ where n is a positive integer. Let C be the set of all bitstrings of length n and let Z be the set of all functions from B to $\{0,1\}$. How do I find the two ...
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3answers
34 views

Clarification regarding function

I have been reading Velleman's How to prove book and this is one of the paragraphs written in the Functions chapter: For every $a \in A$ and $b \in B$, $b = ...
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1answer
35 views

how to define a function?

If we define a function by set theory it states that it is relation in the sets of inputs and outputs such that each input is exactly related to one out put . so if $A=\{9,25,36\} ;\, B=\{3,5,6\}$ ...
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0answers
28 views

A question on function [on hold]

Given $g(y) = \ln(y)$ and $f(y) = \frac{\ln(y^2 + 2y + 1)}{2}$. Show that $g(y) - f(y) < 0$ for $> 0$. What does the question mean by "for $> 0$"?
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1answer
18 views

Notation: Codomain of a probability density function

I need some help with the correct notation for the codomain of a probability density function. Consider the following problem. Let $$ F : V \to (0,1), \, x \mapsto \int\limits_{\inf V}^{x} f(t) \, ...
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3answers
53 views

Limits using definite integration

$F(k)$ = $$ \lim_{n\to \infty}{\frac{1^k + 2^k +...+n^k}{(1^2 + 2^2 +...+n^2)*(1^3 + 2^3 +...+n^3)}} $$ I need help in finding $F(5)$ and $F(6)$. I tried converting it into summation form and using ...
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0answers
35 views

Attaining a maximum without applying the Weierstrauss Theorem

The example that mookid gave in this question is a good one. There is no maximum since it is not continuous. How could you explain why the function given by mookid will attain a maximum on any compact ...
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0answers
26 views

integral of complex function, power series

let $\mu$ be a finite borel measure on $[0,+\infty)$ and let $f$ be defined by $$f(z)=\int_{[0,+\infty)}\frac{d\mu(t)}{t-z},\quad z \in \mathbb{C} \setminus [0,+\infty)\,.$$ *show that the integral ...
6
votes
3answers
147 views

If $f(f(n))=3n$ find $f(2001)$

I have this question which seems a little harder than I thought. It has been about an hour for me hitting aimless thoughts on this one. I can really use a hint here if some one knows how to tackle it. ...
0
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1answer
48 views

Is the Collatz function piecewise linear?

I read somewhere that the Collatz function $\mathbb Z \rightarrow \mathbb Z$: $$\text{Collatz}(x) = \begin{cases} x/2 &&x \; \mathrm{even} \\ 3x+1 &&x \; \mathrm{odd}\end{cases}$$ is ...
0
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1answer
12 views

Definition of a function whose codomain is set of probability measure over cartesian product with dependency between sets in the product

I am thinking about the following function: $$ p : A \to \Delta \big( F(x, y(t) ) \times T \big) ,$$ where $t \in T$ denotes continuous time, and $\Delta (X)$ denotes the set of all probability ...
0
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0answers
26 views

Is this function commonly known or has some name?

I have used this function for fitting in my research, and I wonder if there is a name for it, or is it commonly known in some reduced form? $f(x)=\alpha\frac{e^{-\gamma x}}{x^\beta}+\delta$ Actual ...