Questions tagged [function-and-relation-composition]
For questions about the composition of functions and relations.
0
votes
0answers
8 views
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), (...
4
votes
0answers
80 views
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 ...
1
vote
2answers
35 views
Composite function derivatives
Is this statment right in derivative of compoaite three functions $(f \circ g \circ h)'(x) = (f\circ g)' (h(x))$
0
votes
3answers
18 views
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!
15
votes
1answer
114 views
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 ...
1
vote
4answers
60 views
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 $f(f(x)) = m(mx+b) + b = m^2x + mb + b$, ...
0
votes
1answer
22 views
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 ...
2
votes
1answer
55 views
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: $f_1 = f_2\circ g$? ...
2
votes
2answers
39 views
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 $f(2,3) = 5$ ...
1
vote
1answer
41 views
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 ...
0
votes
0answers
28 views
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)...
1
vote
1answer
30 views
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\...
0
votes
1answer
38 views
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 ...
39
votes
2answers
610 views
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 ...
1
vote
1answer
57 views
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 ...
0
votes
1answer
27 views
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 ...
2
votes
2answers
77 views
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:
...
1
vote
2answers
39 views
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 ...
0
votes
0answers
18 views
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 $$f=f(x,t,s):\mathbb{R}\times [0,+\infty)\...
0
votes
2answers
76 views
What must $f$ satisfy in order to conclude $h \circ f=g \circ f \Rightarrow h = g$
Let $A\left(E,\ E \right)$ be the set of applications from $E$ to $E$ itself.
We equip $A$ with the operation of composition $\circ$.
Question: Which elements $f \in A$ are right regular?
Answer:
$...
0
votes
2answers
50 views
Identity function
I was told that identity functions have the definition when followed.
If f ∘ e = f, e = idx. If e ∘ f = f, e = idy.
The textbook had a question below the definition asking, for any functions f and g,...
2
votes
0answers
26 views
Proof that a sequence of numbers must always contain a perfect square [duplicate]
Given $f(n) = n + \left \lfloor{\sqrt n}\right \rfloor$
Prove that for any natural number $n$, the sequence $n, f(n), f(f(n)),.......$ must always contain at least one perfect square.
Now obviously ...
3
votes
4answers
74 views
Rigorous definition of iterated logarithm
We usually use the recursion theorem to define rigorously the iterated (by composition) $f^n$ of a function,
I want to give a rigorous definition of $\log^{n}x= \log(log^{n-1}x)$ (here there is a ...
4
votes
3answers
395 views
Mapping non-convex functions onto convex functions
I am wondering if a non-convex optimization problem can be reduced to a convex one by mapping non-convex functions/sets onto convex functions/sets. In this context, I would like to know if the ...
1
vote
1answer
30 views
Rank of composition of non-square transpose matrices
Let $d' < d$ be natural numbers, and let $L$ be a real matrix of dimension $d'\times d$, with maximum rank (that is, with rank $d'$). Let $A$ be a real square matrix of dimension $d$, and let's ...
1
vote
1answer
197 views
Unable to understand how to solve piecewise composition of two functions
$$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}...
1
vote
2answers
66 views
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 ...
5
votes
1answer
85 views
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_{...
-1
votes
1answer
24 views
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, ...
0
votes
0answers
43 views
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) $ψ \circ φ (a + b) = ψ \circ φ (a) + ψ \circ φ (b)$ $\forall$ ...
0
votes
2answers
43 views
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
$$h(x)=\begin{cases}f(x)& -1\leq x\leq 0,\\g(x), &\;\;\;\;\;...
1
vote
2answers
36 views
«A cycle is a product of transpositions» $\iff$ «Rearrangement of $n$ objects is the same as successively interchanging pairs»
I am following John B. Fraleigh -- A first course in abstract algebra. On page 90 of the 7th edition he say that
-- Decomposition of a cycle into products of transposition is possible since it is ...
0
votes
1answer
21 views
Composition of Relations (functions)
Working with functions, more specifically function composition. In my case, for starters,
We have an island with the places haven, dale, sun, and ness.
We have a smaller island with the places east, ...
0
votes
1answer
65 views
Do half-iterate functions exist for any function?
For a given a deterministic unary function,
$$ 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 ...
4
votes
2answers
73 views
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 ...
0
votes
2answers
56 views
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-...
3
votes
1answer
85 views
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) = \...
0
votes
1answer
52 views
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 ...
0
votes
0answers
12 views
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 ...
3
votes
1answer
34 views
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 ...
12
votes
2answers
290 views
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 $\...
2
votes
2answers
66 views
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 ...
0
votes
4answers
44 views
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 ...
1
vote
1answer
35 views
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\\ ...
1
vote
1answer
43 views
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 $\{a_0,a_1,...\}$ ...
1
vote
2answers
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 ...
2
votes
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))=...
0
votes
3answers
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 ...
1
vote
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 ...
1
vote
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 ...
