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Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

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Forming a relational composition and justifying its properties

So I have two relationships $Visited \subseteq Student$ X $Country$ $ParentsFrom \subseteq Student$ X $Country$ Visited = { (Mark, Poland), (Mark, UK), (Rick, USA), (Rick, Canada), (Rick, Uganda), (...
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Number of real zeroes of iterated polynomial: $x^3-2x+1$

If $P(x)=x^3-2x+1$, define $z_n$ as the number of real roots of the polynomial $P^{\circ n}(x)$, where the superscript denotes $n$-fold composition. Can we find a general formula for $z_n$, or perhaps ...
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Composite function derivatives

Is this statment right in derivative of compoaite three functions $(f \circ g \circ h)'(x) = (f\circ g)' (h(x))$
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Prohibited values when plugging one formula into another?

Say f(x)=$\frac{1}{x}$ and g(x)=$x-5$ and we where to plug g into f as a composite function, would x = 0 still remain a prohibit value of f or not? Thanks!
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Number of real roots of an iterated quadratic: $x^2-3/2$

I was messing around with polynomials and their real roots when I, as recreational mathematicians do, asked myself the following random question: Suppose I am given a polynomial $P(x)$. How can I ...
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4answers
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All values of $m,b \in \mathbb{R}$ such that $f(x) = mx+b$ is a solution to $f(f(x)) = x$

I am asked to find all values of constants $m,b \in \mathbb{R}$ such that $f:\mathbb{R}\to \mathbb{R}, x\mapsto mx+b$ is a solution to $f(f(x)) = x$. So since $f(f(x)) = m(mx+b) + b = m^2x + mb + b$, ...
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Composition of functions of given set

If ∀x ∈ {1,2,3,4} , $f$(x) = $ x^2 $ and ∀x ∈ {2,4,3,6} , $g$(x) = $ x+1 $. Find $ (g\; o\; f ), (f\; o\; g)\; and \;Im(f \;o \;g)$ I dont understand the question much. If someone can help me it ...
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What is a function $g$ called, which, composed with a certain function $f_2$, yields a given function $f_1$ (i.e. $f_1 = f_2\circ g$)?

Let $X_1$, $X_2$, and $Y$ be non-empty sets. Let $f_1:X_1\rightarrow Y$ and let $f_2:X_2\rightarrow Y$. What's the terminology for a function $g:X_1\rightarrow X_2$ that satisfies: $f_1 = f_2\circ g$? ...
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Is there any unique function $c=f(a, b)$ such that if I know only $c$, I can deduce unique $a$ and $b$ from it? [duplicate]

I want to know a relation $c = f(a, b)$ (where $a, b, c$ are real numbers) such that if I only know $c$ I can deduce $a, b$ inputs from it. As an example consider $f(a, b) = a+b$, then $f(2,3) = 5$ ...
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Decomposing a binary function, into (1) a continuous function and (2) a thresholding function.

Suppose $f$ is an arbitrary binary function and it is given to us: $$ f: \mathbb{R}^{d} \rightarrow \{0, 1\} $$ Prove that it is always possible to write this function, as a composition of two ...
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Seeing the tendency of functions

If $f:[1, 10]\to [1,10]$ is a non decreasing function and $g:[1, 10]\to [1,10]$ is a non increasing function. Let $h(x) =f(g(x)) $ with $h(1)=1$. Then $h(2)$: $\text{(A)}$ lies in $(1,2)$. $\text{(B)...
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1answer
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Function composition - solving for functions

I have a simple, continuous, real-valued function $\sigma$, whose functional form I know, and I know that two other invertible functions, $f$ and $g$, satisfy the following relationships: $f\circ\...
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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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395 views

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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1answer
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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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1answer
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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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1answer
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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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107 views

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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2answers
46 views

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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53 views

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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1answer
78 views

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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1answer
66 views

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 ...