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

learn more… | top users | synonyms (1)

1
vote
2answers
42 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
votes
0answers
20 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
votes
2answers
36 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
votes
1answer
42 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. ...
0
votes
1answer
29 views

How do you symbolically represent the general principle of induction? [on hold]

Normally a specific function is given, and then it would be asked to prove the validity of that specific function with induction. But how do you logically represent the general principle of induction ...
0
votes
2answers
63 views

Periodic function without trigonometry and complex numbers [on hold]

Can I get a periodic function without using trigonometric functions or complex numbers? UPDATE: The question has been superseded by Single-statement Continuous Periodic function without trigonometry ...
3
votes
2answers
68 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 ...
2
votes
2answers
27 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 ...
-1
votes
3answers
50 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
votes
1answer
30 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
votes
1answer
18 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 ...
1
vote
1answer
16 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 ...
-1
votes
3answers
31 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
votes
2answers
28 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 $x\in[0,1]$?
0
votes
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
votes
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
votes
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
votes
1answer
50 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
votes
0answers
21 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
votes
0answers
23 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
votes
4answers
219 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
votes
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 ...
-1
votes
2answers
25 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 ...
0
votes
5answers
62 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
vote
2answers
64 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
vote
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. ...
0
votes
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 ...
0
votes
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 ...
1
vote
4answers
55 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
votes
1answer
34 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
162 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
votes
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
votes
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
votes
1answer
37 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
votes
2answers
46 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
31 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 ...
-1
votes
1answer
72 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 ...
0
votes
3answers
35 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 = ...
1
vote
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 $A=\{9,25,36\} ;\, B=\{3,5,6\}$ ...
0
votes
0answers
41 views

A question on function [closed]

So,I took this question from my friend assignment which he got from his teacher,I had no idea about the question as it use natural log in function.The question goes: Given $g(y) = \ln y$ and $f(y) = ...
0
votes
1answer
19 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) \, ...
1
vote
3answers
55 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 ...
0
votes
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 ...
0
votes
0answers
31 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
151 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
votes
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
votes
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
votes
0answers
27 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 ...
0
votes
0answers
18 views

Understanding the meaning behind plotting a “gradient vector” on a graph containing contour lines?

A rather basic question here, please do forgive any technical errors in the question. Throughout this example consider the general function w=f(x,y) I am used to visualizing derivatives the same way ...