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Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

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What are examples of pseudorandom functions that can be preformed without the assistance of a computer?

I'm trying to generate a seemingly random list of integers without the use of a computational device; one capable of being unraveled with a mathematical function. What's the best way to devise a ...
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Which functions satisfy $f^n(x) = f(x)^n$ for some $n \ge 2$?

Let $n$ be an integer greater than $1$. The notation $f^n$ is notoriously ambiguous: it means either the $n$-th iterate of $f$ or its $n$-th power. I was wondering when the two interpretations are in ...
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Is the Schröder Equation valid for higher dimensional iterated maps?

The Schröder's functional equation is the eigenfunction equation for the composition operator given as: $$ \psi \circ y (x) = s \cdot \psi(x) ~~~~~~~~~~~ (1) $$ The interesting bit about it (at ...
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Structure of Proofs of Invertibility?

When trying to prove the invertibility of an objects (say, the composition of two functions), do we always structure the proofs by showing, not necessarily in this order, (1) that the inverse of the ...
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What is the correct way to read $f\circ g$?

Let $f : X \to Y$ and $g : Y \to Z$ be functions. We define the composition $g \circ f : X \to Z$ by $g \circ f(x) = g(f(x))$ for each $x \in X$. I have also heard the composition read out like this: ...
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Restrictions of compositions

Considering two functions, $f,g:X\to X$ s.t. $Y\subseteq X$ is invariant for both $f$ and $g$, that is $f(Y)\subseteq Y$ and similar for $g$. Do we then have $f|_Y\circ g|_Y=(f\circ g)|_Y$. It seems ...
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Partial derivative of a composite function - weak conditions

Let $h=h(u,w):\mathbb{R}\times [0,+\infty)\longmapsto \mathbb{R}$ be a function such that $h,h_u\in C(\mathbb{R}\times [0,+\infty))$. Consider the function $$f=f(x,t,s):\mathbb{R}\times [0,+\infty)\...
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What must $f$ satisfy in order to conclude $h \circ f=g \circ f \Rightarrow h = g$

Let $A\left(E,\ E \right)$ be the set of applications from $E$ to $E$ itself. We equip $A$ with the operation of composition $\circ$. Question: Which elements $f \in A$ are right regular? Answer: $...
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Identity function

I was told that identity functions have the definition when followed. If f ∘ e = f, e = idx. If e ∘ f = f, e = idy. The textbook had a question below the definition asking, for any functions f and g,...
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Proof that a sequence of numbers must always contain a perfect square [duplicate]

Given $f(n) = n + \left \lfloor{\sqrt n}\right \rfloor$ Prove that for any natural number $n$, the sequence $n, f(n), f(f(n)),.......$ must always contain at least one perfect square. Now obviously ...
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Rigorous definition of iterated logarithm

We usually use the recursion theorem to define rigorously the iterated (by composition) $f^n$ of a function, I want to give a rigorous definition of $\log^{n}x= \log(log^{n-1}x)$ (here there is a ...
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Mapping non-convex functions onto convex functions

I am wondering if a non-convex optimization problem can be reduced to a convex one by mapping non-convex functions/sets onto convex functions/sets. In this context, I would like to know if the ...
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Rank of composition of non-square transpose matrices

Let $d' < d$ be natural numbers, and let $L$ be a real matrix of dimension $d'\times d$, with maximum rank (that is, with rank $d'$). Let $A$ be a real square matrix of dimension $d$, and let's ...
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Unable to understand how to solve piecewise composition of two functions

$$f(x)= \begin{cases} 2x & ; x < 0 \\ \sqrt{x} & ; 0 \leq x \leq 1 \\ (x-1)^2+1 & ; x > 1 \end{cases} $$ $$g(x)= \begin{cases} x^2 & ; x \leq 1 \\ 1 &; x > 1 \end{cases}...
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Limit of composite functions

Let $f$ and $g$ be some functions, assuming all right conditions that allow function composition, I want to prove that $$\lim_{x \to \infty} f(g(x))=f\left(\lim_{x \to \infty}g(x)\right) $$ As long ...
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Is there a standard notation for the pre-composition operator?

Let $X_1$, $X_2$, and $V$ be sets. Is there a standard name and a standard notation for the pre-composition operator $F$ that takes as input a function $\varphi:X_2^{X_1}$ and returns the operator $F_{...
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A question about finding the inverse of a function

Question: $f : \mathbb Z\times\mathbb Z\to\mathbb Z\times\mathbb Z$ is defined by $f((x, y)) = (y, x)$. Write down whether $f^{-1}$ exists. If it does, write down $f^{-1}((3, 4))$. If it doesn’t, ...
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If $φ: A → B$ and $ψ: B → C$ are homomorphisms, then $ψ \circ φ: A → C$ is a homomorphism.

Being $A, B$ and $C$ rings. To prove the above statement, I need to show that $ ψ \circ φ: A → C $ satisfies the following conditions: i) $ψ \circ φ (a + b) = ψ \circ φ (a) + ψ \circ φ (b)$ $\forall$ ...
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Is this composition of functions $h(x)=\begin{cases}f(x)& -1\leq x\leq 0,\\g(x), &\;\;\;\;\;\;\;\;\;x>0\end{cases}$ continuous?

Given that $f:[-1,0]\to\Bbb{R},\;\;g:[0,1]\to\Bbb{R},$ $f,g$ continuous with $f(0)=g(0).$ Let $h:[-1,1]\to\Bbb{R}$ be defined as $$h(x)=\begin{cases}f(x)& -1\leq x\leq 0,\\g(x), &\;\;\;\;\;...
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«A cycle is a product of transpositions» $\iff$ «Rearrangement of $n$ objects is the same as successively interchanging pairs»

I am following John B. Fraleigh -- A first course in abstract algebra. On page 90 of the 7th edition he say that -- Decomposition of a cycle into products of transposition is possible since it is ...
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Composition of Relations (functions)

Working with functions, more specifically function composition. In my case, for starters, We have an island with the places haven, dale, sun, and ness. We have a smaller island with the places east, ...
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Do half-iterate functions exist for any function?

For a given a deterministic unary function, $$ f(x) = y $$ A half-iterate function g(x) for f(x) is one which satisfies the following: $$ g(g(x)) = f(x) $$ For any f(x), does there exist a ...
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No. of boolean functions of $n$ variables constructible from AND and OR, but without NOT

Consider a Boolean function $f(x_1, x_2, \dots, x_n)$ from $\{0,1\}^n$ to $\{0,1\}$. It is well known that there are $2^{(2^n)}$ such functions, because for each of $2^n$ possible input vectors you ...
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Functions which are the identity when applied repeatedly

Clearly the identity function $f =\text{id}$ satisfies $f^n = \text{id}$ for any $n \in \mathbb{N}$. However, there are also other functions with this property. For instance, with $n=2$ we have self-...
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What causes a set of functions to form a group under composition?

In my textbook the following functions are given and are said to form a group under composition: $f_1(x) = x$, $f_2(x) = \frac{1}{x}$, $f_3(x) = 1-x$, $f_4(x) = \frac{1}{1-x}$, $f_5(x) = \...
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Correct way to formally prove whether a composite function exists or does not exist

I am confident that my answer is correct, but I am not sure if the manner in which I have proved it is formal enough. g(-1,-2) = (1/2 * 1, -2) = (1/2,-2) x = 1/2 is within the parameter of the ...
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Is the multi-dimensional Gevrey-class $G^1$ closed under composition?

Let $d,n \in \mathbb{N}$, and let $\Omega \subseteq \mathbb{R}^d$ and $\Psi \in \mathbb{R}^n$. We say that $f \in G^1(\Omega,\Psi)$ if all of its coordinate functions $f_1, \dots, f_n$ are elements of ...
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Function composition in $L^2$

Let $f\in L^2(0,\infty)$ with $|f(x)| \leq |x|$. Further, define $g(x)=d^x$ for some $d>1$. Question: Is $f\circ g \in L^2(\mathbb{R})$? If yes, how do I show this? If no, under which conditions ...
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What numbers are in the set $S$?

Here is a problem that I came up with that has been bothering me for a while, and I haven't been able to solve it. Let $S$ be the set defined by the following rules: $2\in S$. Define the functions $\...
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Why does continuity of the composite function $f\circ g$ at $c$ require continuity of the function $g$ at $c$?

I'll state a question from my textbook below: Given $f(x) = \frac 1 {x-1}$. Find the points of discontinuity of the composite function $y = f[f(x)]$. Clearly, $f(x)$ is not defined at $x=1$. But ...
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What is the domain of this composite function?

The question is: $f(x) = \dfrac{x}{x-1}$ $g(x) = \dfrac{1}{x}$ $h(x) = x^2 - 1$ Find $f \circ g \circ h$ and state its domain. The answer the textbook states is that the domain is all real values ...
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Composition of $h(x)=f(|g(x)|)+|f(g(x))|$

$f(x)=\begin{cases}x-1,\ &-1\leq x\leq0\\ x^2,\ &0< x\leq1\end{cases}$ and $g(x)=\sin x$. Find $h(x)=f(|g(x)|)+|f(g(x))|$. $$f(|g(x)|)=\begin{cases}|\sin x|-1,\ &-1\leq |\sin x|\leq0\\ ...
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What is the canonical name (or a good one) for the following self-composition of functions?

In trying to answer (my own) recent question I'm looking for the name for the following type of self-composition of a function. Let's define an infinite set of fixed coefficients $\{a_0,a_1,...\}$ ...
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A “half-iterate” of the Faulhaber-problem of “summing like powers”?

Background: In a question about the "sum of sums of $k$th powers of first natural numbers" someone asked -in principle- for the 2'nd iteration of the Faulhaber-problem. The Faulhaber's problem can be ...
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Roots of composite polynomial function

For a polynomial function a(x), is there a generalized solution for the roots of: $$a(a_2(a_3(...(a_m(x))...)))$$ As an example, if: $$a(x)=x^2-5x+2, m=2$$ How would I find the roots of $$a(a(x))=...
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If $g \circ f$ is injective and $f$ is surjective, then $g$ is injective. [duplicate]

Let $f : A \to B$ and $g : B \to C$ be functions. If $g \circ f$ is injective and $f$ is surjective, then $g$ is injective. This question that correctly proves this theorem, but my proof seems ...
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Is there a short notation for function composition?

So I think that if there exists two functions $f_1 \colon X \to Y$ and $f_2 \colon Y \to Z$ you can notate their composition using $(f_1 \circ f_2)(x) = f_3(x)$ right? If I have many functions that I ...
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Checking if a mapping exists in a composition of partial functions

First of all apologies because I'm not a mathematician. So let's suppose I have a number of sets $A_1,...,A_n$ and functions $f_i\colon A_i \to A_{i+1}$ with $1\le i<n$. I want to know how to ...
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which is the function of (3) based in this equation

If we know that $(f\circ f)(x)=4x+3$, with $f(0)=4$, what is $f(3)=?$ I have found that $f(f(x))= 16x+15$, but I don't know where to go from there.
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Doubt in composite function derivative

Consider this function: $$f(x) = 2x +1$$ it can be seen as a composite function: $f(g(x))$ with $$f(x) = x + 1$$ $$g(x) = 2x$$ Using the chain-rule to derive the original $f(x)$ I got: $$f'(x) = ...
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restriction on the domain of composition of functions

In a set theory class I was taught composition of functions as a special case of composition of relations, and to think of dom $g\circ f$ as dom $f \cap f^{-1}[$dom $g]$. Then when we speak of ...
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An Identity for a Fibbonacci-Type Polynomial

Problem: The polynomials $p_{n}\left(x\right)$ are defined recursively by the linear homogenous order 2 difference equation $$p_{n+1}\left(x\right)=2\left(1-2x\right)p_{n}\left(x\right)-p_{n-1}\left(...
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How many composite functions from $(f \circ f)(1) = 2$

Let $S = \{1,2,3,4\}$. Let $F$ be sets of all functions from $S$ to $S$. How many $f\in F$ are there so that $(f \circ f)(1) = 2$ ? How I see it, is through a diagram and so that there is $4*4*4*4$ ...
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Am I on the right track to proving this statement about composite function?

Let $S = \{1,2,3,4\}$. Let $F$ be the sets of all functions from $S$ to $S$. Now I think this statement is true: $\forall f \in F , \exists g\in F$ so that $(g \circ f)(1) =2$ I suppose $f \in F$, ...
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Composite functions, is this statement correct? [duplicate]

Let $S = \{1,2,3,4\}$. Let $f$ be the sets of all functions from $S$ to $S$. Now, is this statement correct? $\forall f \in F$ , $\exists$ $g \in F$ so that $(f\circ g) (1) = 2$ I think this is ...
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1answer
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Are the following either one-to-one or onto functions?

I just want to see if i'm on the right path in determining if the following are onto or one to one. $f\circ g = 3 \lfloor (x+1)/2 \rfloor$ $g\circ f = \lfloor (3x+1)/2 \rfloor$ Both functions are ...
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1answer
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Not sure if this statement is true - Composition functions

$\forall$ functions $f,g,h$ if $f \circ h = g \circ h$ and $h$ is onto, then $f=g$ . I think this statement is true, as I draw diagrams for myself. However I have trouble proving it so I don't know ...
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Graphs in a regular category

Let $\mathcal{C}$ be a regular category, and let $X,Y,Z\in Ob(\mathcal{C})$. Let $g:Z\to Y$ be a regular epi, and let $R\in Sub(X\times Y)$ (subobjects of $X\times Y$). Define $S:=(id_X\times g)^\ast(...
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Is this proof about composition functions correct?

For this to work i did draw diagrams and i came up with example of functions. Let F be the sets of all functions from the set of integers to set of integers. Prove or disapprove: FALSE $\forall f,...
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Relation for more than 2 sets.

Do we have some concept of relation between more than two sets (composition of relations) as we have definition of Cartesian product for $n$ sets and composition of function. What I intend to ask is ...