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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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Fully spanning subset

Consider a finite multiset of integers denoted by M. I define a subset of that multiset denoted by A. I say that A is a Fully Spanning Sub-set of M if, for every item x in M, if ...
JoeHills's user avatar
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0 answers
30 views

A surjective continuous open map has a continuous right inverse [duplicate]

My question arises from this post and similar questions. It's clear that for an open, onto and continuous function $f:X\rightarrow Y$, not every right inverse is continuous, but my question is if ...
H4z3's user avatar
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4 votes
1 answer
111 views

Solutions to $(f(x)-f(y))^3=f\left(x^3\right)-f\left(y^3\right)$

I was wondering, if there are more solutions to the functional equations, than $f(x) = const$. Maybe someone has an idea of how to find all solutions (or all continuous solutions)? Find all the ...
Vlad Boiko's user avatar
1 vote
0 answers
55 views

Finding a function in the unit sphere of a functional subspace with a couple of properties

Preliminaries: A={f $\in C(X); f(a)=0$} is a banach space with norm the following: $\Vert f\Vert=sup\vert f(x)-f(y)\vert; x,y \in X$ ( X is Hausdorff and compact space. Element of a is in X. ...
Oushin Tanakara's user avatar
-3 votes
1 answer
45 views

Limit of a function when 'a' is not in the domain [closed]

Is this a correct statement that as x approaches to 'a' for f(x) where 'a' does not belong to the domain of f(x) then the limit at 'a' does not exist
Rit Mukherjee 's user avatar
0 votes
1 answer
54 views

Closed form for the area under $ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $

Define a function $f:\Bbb Q \to \Bbb Q$ by the following $$ f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)} $$ where $\pi(\cdot)$ is the prime counting function and $N\in \Bbb N.$ I would like to find ...
zeta space's user avatar
19 votes
9 answers
2k views

Formula for bump function

I would like to formulate a bump function (link) $f(x)$ with the following properties on the reals: $$ f(x) := \begin{cases} 0, & \mbox{if } x \le -1 \\ 1, & \mbox{if } x = 0 \\ 0, & \...
Richard Burke-Ward's user avatar
-2 votes
1 answer
106 views

Is there an analog for factorials in division, and if so, what are its applications and properties? [closed]

If we consider a factorial to be an operation/function of iterative multiplication, would it be reasonable to think that something similar for division also exists? If we take this function to be f, ...
Pratixit Tripathy's user avatar
0 votes
0 answers
18 views

How to find a measurable, Lebesgue invariant bijection between the interval and the 3-dimensional sphere [closed]

From a book I was reading I have the following statement: "It is a general fact of measure theory that there is a bijection $f : [0, 1) → \mathbb{S}^2$ such that both $f$ and $f^{-1}$ take ...
Riel Blakcori's user avatar
0 votes
1 answer
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Given a functional equality prove that the function is never $0$

For a function $u:\mathbb{R} \to\mathbb{R}$ with $u(t)^2+u(t)=t^2+t+2$ , $u(0)=1$, Can we prove that the graph of $u$ never touches the $t$ axis ? We can easly see that by adding $0.25$ to both sides ...
Antony Theo.'s user avatar
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-5 votes
0 answers
24 views

With certain conditions, what are the properties of function f? [closed]

The function $f(x)$ is defined on the set of real numbers in such a way that we have : $f(x)f(x+1) - f(f(x+1)) = f(2x)$ a) What general properties should the function $f(x)$ have? b) Can the function $...
زهرا میرمعصومی's user avatar
-5 votes
2 answers
58 views

Why can't I find the range of the function $\frac{1}{\sqrt{x-5}}$?

$f(x) = \frac{1}{\sqrt{x-5}}$. Now $f(x) = y$; and $x = g(y)$. Range of $f(x)$ = Domain of $g(y)$ $y = \frac{1}{\sqrt{x-5}}$ $y^2 = \frac{1}{{x-5}}$ $x = \frac {1 + 5y^2}{y^2}$ Clearly, the domain of ...
Math Math Math's user avatar
1 vote
0 answers
23 views

Interpreting roots of multiplicity greater than 1 in characteristic polynomial in generating functions

This might be a stupid problem... In Wilf's Generatingfunctionology he proposed a method to solve the following "initial value problem" $$ (a+bx+cx^2)U(x) = x(D(x)+au_1+cu_{N-1}x^N) $$ $a, b,...
Haimu Wang's user avatar
0 votes
1 answer
24 views

Characterize a specific measurable space

This exercise is from J.F Le Gall GTM294, Measure theory, probability and stochastic processes. Exercise 4.8 (1) Let $f$ and $g$ be two nonnegative measurable functions on a measure space $(E,\mathcal{...
Stellaria's user avatar
0 votes
1 answer
93 views

Determine all possible values ​of the sum $S=a+b+c+d$

The problem Let $a$ and $b$ be two integers for which the interval $(a,b]$ contains 20 integers and let $c,d$ be two natural numbers for which the interval $(c,d)$ contains 24 natural numbers. ...
IONELA BUCIU's user avatar
0 votes
0 answers
28 views

Single Crossing Property - How to prove Neatly

I have the following problem, and want to learn how to proceed in a very precise and short manner to show the Strict Single Crossing Property (SSCP) for $U(.)$ without the brute force approach. I ...
Pluviaum's user avatar
3 votes
1 answer
49 views

How to map identity to a sigmoid?

Is there a way of smoothly defining a function that transforms the identity function to a sigmoid for a fixed range (say $[0,1]$)? What I want is to define a function $f(x,k)$ such that $f(0,k)=0,f(0....
sam wolfe's user avatar
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2 votes
1 answer
77 views

Do $\operatorname{arcsec}\left(\frac{2x}{5x+3}\right)$ and $\operatorname{arccos}\left(\frac{5x+3}{2x}\right)$ have the same domain?

Do $$f(x)=\operatorname{arcsec}\left(\frac{2x}{5x+3}\right) \quad\text{and}\quad g(x) = \operatorname{arccos}\left(\frac{5x+3}{2x}\right)$$ have the same domain? I wondered about this while finding ...
improvement dude's user avatar
0 votes
0 answers
11 views

Is there an exact expression for the full width half maximum of a sech^2 curve convolved on itself?

As some simple math can show, a Gaussian convolved onto itself is also a Gaussian. Importantly, the FWHM of the original gaussian compared to that of its convolved counterpart is different by a factor ...
ChemGuy's user avatar
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-1 votes
4 answers
91 views

How to find the range of a quadratic function we can't use the quadratic formula?

In the question $f(x) = \frac {x}{(1+x^2)}$ $ yx^2 - x + y = 0 $ $x = \frac {1 ± \sqrt {1-4y^2}}{2y}$ We know that $x$ belongs to $\Bbb R$. So, $\frac {1 ± \sqrt {1-4y^2}}{2y}$ also belong to $\Bbb R$....
The's user avatar
  • 1
3 votes
0 answers
88 views

Is there a function whose maximizers remain the same after any affine transformations?

Let $f: \mathbb{R_+}^n\to \mathbb{R_+}$ be a function that is strictly increasing in each of its arguments. Let $M_f$ be the set of its maximizers on some fixed compact subset $D\subseteq \mathbb{R_+}^...
Erel Segal-Halevi's user avatar
-3 votes
0 answers
25 views

Relationship between functions (operators). [closed]

Basically almost every mathematical phenomenon which we call it is due the result of a particular function, can be defined using the combination of basic functions that are related to each other in ...
Sanskar Anand's user avatar
0 votes
0 answers
24 views

Independent Variable Technique in Derivative of a function with respect to another function

This question differentiates $y=x$ with respect to $x^2$ by introducing a variable $u$, and answers say his method is valid. On the other hand, the same technique when used here has been downvoted, ...
Starlight's user avatar
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1 vote
2 answers
65 views

Why is interpolating $y=g(x)$ then applying $h(y)$ not equivalent to interpolating $h(g(x))$?

Say I have a table of voltage, current, and resistance values as so. V [V] I [A] R [$\Omega$] 1 2 0.5 3 5 0.6 The V and I columns are measurements, R is a simple calculation from Ohm's law (V=IR). ...
jrecord's user avatar
  • 13
0 votes
1 answer
57 views

Prove that a functional equation has at least one root

Let $f$ be a strictly increasing and continuous function that satisfies the equality $$f^3(x)+βf^2(x)+γf(x)=x^3-2x^2+6x-1, \forall x\in\mathbb{R} $$ with $β,γ\in\mathbb{R}$ s.t $β^2<3γ$ Prove that ...
Antony Theo.'s user avatar
  • 1,530
2 votes
0 answers
47 views

Alternate proof to the Extreme Value Theorem

I'm following Spivak's Calculus and was revisiting some of my notes when I think I found a much more straightforward proof for the Extreme Value Theorem, compared to the one given in the book. I was ...
Aryaan's user avatar
  • 283
1 vote
1 answer
43 views

How to write this in a rigorous way? [duplicate]

I was studying for a test and encountered the following problem in a textbook: Suppose $f(x)$ is continuous, with $f >0$ for all $x$ and $\lim_{x \to \infty} f(x) = \lim_{x \to -\infty}f(x)= 0$ ...
Artur Stolf's user avatar
4 votes
3 answers
131 views

Find minimum value of $f(x)= x^{1.5} + x^{-1.5} -4(x + x^{-1})$

Problem: Find minimum value of $f(x)= x^{1.5} + x^{-1.5} -4(x + x^{-1})$ My first attempt involved letting $\lambda = \sqrt{x}+\frac{1}{\sqrt{x}}$ followed by successive squaring and cubing and then ...
Vansh Chandak's user avatar
0 votes
0 answers
13 views

How to constrain a general mapping (from a multiset to a multiset) to a multirelation?

A multiset is a function, that assign a multiplicity (non-negative integer) to all elements of an underlying (universe) set. Let $A$ be a set, then $m: A \to \mathbb N_0$ is a multiset. $\mu A$ ...
Minop's user avatar
  • 101
0 votes
0 answers
15 views

Complete Lattices and the Injectivity of the Restriction $f|_S$ - Verification of Proof

Attempt (General Case) Conjecture: I want to show that if $X$ and $Y$ are nonempty sets, $(X, \leq)$ is a complete lattice, and $f: X \to Y$ is any well-defined function, then there exists a nonempty ...
Joshua Ortiz's user avatar
0 votes
0 answers
41 views

Closed form solution of equation by finding a suitable function

Starting with a sum such as $\sum_{i} b_{i}$, where $b_{i} > 0$ are real numbers for all $i$ under consideration, I have a corresponding vector of real numbers $a_{i} > 0$ for all $i$. I want to ...
Thomas Fjærvik's user avatar
0 votes
1 answer
57 views

(How) can I determine the point of tangency of $k\cdot x$ and $\sin{x}$?

Is it possible to determine the exact coordinates of the point of tangency of a line through the origin $k\cdot x$ and $\sin{x}$, with a $k$ chosen so there exists a unique point of tangency? The ...
MaxD's user avatar
  • 866
0 votes
1 answer
90 views

Does there exist a widely-used operator $\boxdot$ such that $(\theta \boxdot \phi)(x) := \theta(x) \circ \phi(x)$?

Let $\forall X,Y : L(X,Y)$ symbolize the set of all linear operators from $X \rightarrow Y$. Let us have operator-valued functions $\theta : I \rightarrow L(Y,Z)$ and $\phi : I \rightarrow L(X,Y)$. It ...
Timothy Leong's user avatar
1 vote
2 answers
43 views

Solving for the range of a function algebraically

I am trying to find the range of: $$ \frac{x}{x^2-16} $$ I first solved for y: $$ y= \frac{x}{x^2-16} $$ $$ y(x^2 -16) = x $$ $$ yx^2 -16y = x $$ $$ yx^2 -x -16y = 0 $$ I then used the quadratic ...
Matx's user avatar
  • 43
-1 votes
1 answer
94 views

Solutions to $y'=y$

This is a pretty famous ODE. Solution $$\frac{dy}{dx}=y\implies \frac{dy}{y}=dx$$ Integrate both sides and we get $$\ln y=x+C \implies y=C_1e^x$$ My question Why is $y=0$ not a valid solution? $C_1=e^...
Gwen's user avatar
  • 2,729
1 vote
1 answer
55 views

How to say these two distinct functions have the same structure?

Yesterday, I posted this question, which remains unanswered. In this related question, I ask a different yet more precise question that may help me solve the other question. Let $N=\{1,2\}$ be a two-...
EoDmnFOr3q's user avatar
  • 1,226
0 votes
1 answer
54 views

Prove that assuming $f:S\rightarrow T$, $f$ is a bijection iff there is $g:T\rightarrow S$ such that $f\circ g$ and $g\circ f$ are identity maps

I'm trying to prove the following: Let $S$ and $T$ be sets and $f: S \rightarrow T$. Show that $f$ is a bijection iff there is a mapping $g: T \rightarrow S$ such that $f \circ g$ and $g \circ f$ are ...
Ali's user avatar
  • 356
6 votes
3 answers
359 views

What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?

We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$ is $n$ (thanks to this question), but what is the global maxima of $\frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}...
RajaKrishnappa's user avatar
1 vote
0 answers
120 views

An interesting bijective function on $\Bbb Z_{26}$

In order to get the last letter of any italian tax code it is used the following bijective function on $\Bbb Z_{26}$. $f:\Bbb Z_{26}\to\Bbb Z_{26}$ defined as follows : $f(0)\!=\!1,\,f(1)\!=\!0,\,f(2)\...
Angelo's user avatar
  • 12.5k
0 votes
0 answers
37 views

Proving there exist $g,h$ where $g = \Theta(h)$ and $f(x) = g(x) - h(x)$ for a function $f$

I am trying to prove that for any function $f : \mathbb{Z}^{+}\rightarrow \mathbb{R}^{+}$, there exist $2$ functions $g : \mathbb{Z}^{+}\rightarrow \mathbb{R}^{+}$ and $h : \mathbb{Z}^{+}\rightarrow \...
Princess Mia's user avatar
  • 3,019
2 votes
0 answers
49 views

Function definition estimation

Good day, not a mathematician here! I have been trying to approximate a certain function given some known and unknown values. Let $ M $ be a variable such that $M$ ranges from 1 to 106 with a step ...
Vladouch's user avatar
1 vote
0 answers
63 views

Continous function with infinitely many zeroes

Let's say I have a function $f(x)$, that is continous everywhere on the real line. Suppose I have a sequence of real numbers $a_k$ such that $a_0$ = $C$ and as $k$ tends to infinity, $a_k$ tends to a ...
Riccardo Caiulo's user avatar
2 votes
6 answers
231 views

Maximum value of $2$ variable function $f(u,v)=\frac{\left(1-\sqrt{uv}\right)^2}{\frac{1-u^2}{2u}+\frac{1-v^2}{2v}}$

Finding maximum value of $\displaystyle f(a,b)=\frac{\bigg(1-\sqrt{\tan\frac{a}{2}\tan\frac{b}{2}}\bigg)^2}{\cot a+\cot b}$, Where $a,b\in\bigg(0,\frac{\pi}{2}\bigg)$ What I try : $\displaystyle \...
jacky's user avatar
  • 5,200
1 vote
0 answers
38 views

Any algebraic simplification for the following?

As part of a problem I'm solving with polynomials I am confronted with the expression: $$f(x,y) = \left(\prod_{m=0, m\neq j}^na_j-a_m\right)^{-1}\left(\prod_{m=0, m\neq j}^n(x+y-a_m) - \prod_{m=0, m\...
MokutekiJ's user avatar
  • 166
0 votes
1 answer
29 views

Derivation of Legendre Polynomials from only orthogonality

I recently stumbled on the idea of Legendre polynomials, from the perspective of using them to approximate functions over a region, and I discovered all of these other ways to express them, using ...
Thomas Blok's user avatar
0 votes
1 answer
74 views

Determining $t(x)$ from $\frac{dx}{dt}$?

I have a question that "feels" basic/stupid, but I've been really struggling with it. The basic question is: is there an easy way to determine $t(x)$ from $x(t)$ or $\frac{dx}{dt}$? To ...
UserHandel's user avatar
0 votes
1 answer
54 views

Proving $f + c = O(f)$ doesn't always hold- where is my mistake?

I seem to have proved the following statement false: that for any function $f : \mathbb{Z}^{+}\rightarrow \mathbb{R}^{+}$ and any $c \in \mathbb{R}, f +c = O(f)$, where for any $2$ functions $f : \...
Princess Mia's user avatar
  • 3,019
1 vote
0 answers
29 views

Odd Function/Integral when evaluating a Moment

I am studying the Laplace (Double Exponential) distribution, and I have the following quote from Siegrist, which is quite direct, and not bothered about conditions being fulfilled: That the odd order ...
Starlight's user avatar
  • 1,834
0 votes
0 answers
46 views

How to find period of an arbitrary periodic function?

Is there any known algorithm to find the period of a generic function which is known to be periodic? The most direct approach would be to solve for $T$ the functional equation $$f(x+T)=f(x)$$ which is ...
Sanjana's user avatar
  • 265
0 votes
2 answers
70 views

Let $p(x)$ be a polynomial and $f(x)$ a line that is tangent to $p(x)$ at $r.$ Why must $p(x)-f(x)$ have a root of $r$ with even multiplicity?

I recently came across a problem which needed the fact that if $p(x)$ is a polynomial and $f(x)$ is a line with $f(x)$ being tangent to $p(x)$ at point $r$, then the function $p(x)-f(x)$ must have a ...
babyaids's user avatar

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