Questions tagged [function-and-relation-composition]
For questions about the composition of functions and relations.
1,174
questions
0
votes
1
answer
54
views
How to understand such a function $g(I+X)$?
Recently I've been reading Lie Groups written by Daniel Bump.
Lemma 7.1. Let $f$ be a smooth map from a neighborhood of the origin in $\mathbb{R}^n$ into a finite-dimensional vector space. We
may ...
- 93
0
votes
0
answers
16
views
Inverse of composition of functions with intermediate change in dimension
Consider the simple function below:
$$y=f(x)=Ag(Bx)$$
where $x\in\mathbb{R}^n$, $y =f(x)\in\mathbb{R}^n$, $B\in\mathbb{R}^{m\times n}$, $A\in\mathbb{R}^{n\times m}$, and an invertible $g(\cdot): \...
- 150
1
vote
2
answers
80
views
Showing that for all $f: X \to Y$ there exists $S, g: X \to S, h: S \to Y$ with $g$ injective and $h$ surjective such that $f = h \circ g$
I want to show the claim in the title: That for all functions $f: X \to Y$ there exists some set $S$, some injective function $g: X \to S$, and some surjective function $h: S \to Y$ such that $f = h \...
1
vote
0
answers
40
views
Showing that for all $f: X \to Y$ there exists $S, g: X \to S, h: S \to Y$ with $g$ surjective and $h$ injective such that $f = h \circ g$ [duplicate]
I want to show the claim in the title: That for all functions $f: X \to Y$ there exists some set $S$, some surjective function $g: X \to S$, and some injective function $h: S \to Y$ such that $f = h \...
1
vote
0
answers
29
views
Domain and range of an inverse trigonometric composite function.
arcsin([x]/{x}) - what will the domain and range of such a function be?Here, [x] implies the greatest integer function while {x} is the fractional part function.
I tried using the fact that for arcsin(...
1
vote
0
answers
21
views
Distortion Lemma for composition of (distinct) functions expanding on average
I am trying to describe the following dynamics:
Let $(T_{\rho})_{\rho \in [0, 1]}$, $T_{\rho}: [-1, 1] \rightarrow [-1, 1]$ be a family map which satisfies:
$\forall \, \rho \in [0, 1], \, \exists \,...
-3
votes
0
answers
35
views
$f$ and $g$ are functions from $\mathbb{R} \to \mathbb{R}$. If $f(x) = 2x+6$ and $g(x)= x^3$, what is $g\circ f$?
I was able to get the first part of this question which was $f\circ g = 2x^3+6$. I was unable however to answer the question which I posted.
- 11
0
votes
1
answer
34
views
Confusion regarding the definition of composition of functions
This question may seem kinda silly but in constructing a well organized proof about the associativity of function compositions I need to clear my confusion.
Here's the definition of composition of ...
1
vote
0
answers
54
views
This function has a fixed point, $F^4(x)=x$. Why?
Consider two points on the unit sphere, $C_1$ and $C_2$. Let these points be close enough such that circles of radius $r$ drawn around each point intersect at two points, $N_1$ and $N_2$. The vectors ...
- 127
-1
votes
1
answer
57
views
Composition of $H^2$ functions [closed]
Suppose $\Omega$ is a sufficiently smooth bounded domain of $\mathbb{R}^2$. Let $f \in H^2(\Omega) = W^{2,2}(\Omega)$ and $g \in H^2(\Omega) \cap L^{\infty}(\Omega)$.
Can we say that $g \circ f \in H^...
- 75
0
votes
1
answer
28
views
If $T ◦ R$ and $T ◦ S$ are disjoint, then so are $R$ and $S$.
Suppose $R$ and $S$ are relations from $A$ to $B$ and $T$ is a relation from $B$ to $C$. If $T ◦ R$ and $T ◦ S$ are disjoint, then so are $R$ and $S$.
I am having trouble proving it (and so far, I ...
- 53
2
votes
3
answers
60
views
Is the domain of $g(f(x))$ always a subset of domain of $g(x)$?
Let there be two functions $f(x)=\sqrt{x}$ and $g(x)=\sqrt{2-x}$.
So, $g(f(x))=\sqrt{2-\sqrt{x}}$.
As evident the domain of $g(x)$ is $(-\infty, 2]$ and the domain of $g(f(x))$ is $[0,4]$. But from ...
- 75
4
votes
0
answers
123
views
Under what conditions is $\lim_{x\to a}\left|\varphi\circ f(x)-\tau \circ g(x)\right|=0$ true?
This question is inspired from another much easier problem I was trying to solve which I tried to generalize. The question is essentially as follows (assuming all the limits exist)
If $a\in \mathbb R\...
- 8,122
0
votes
0
answers
41
views
systems of equations involving composition of functions
If we are given a function
$g(x) = x - 1/x$
And another one given in terms of composition
$f(g(x)) = x^3 - 1/x^3$
By which general method does one find $f(x)$ ?
Can it be found for arbitrary $g(x)$ ...
- 133
2
votes
2
answers
78
views
Why codomain is more than the range in an Inverse function
While solving inverse function problems, I got confused in a part, like for any Inverse function to be defined, it must be one-one and onto, then in many questions why the codomain is given more than ...
- 23
0
votes
1
answer
85
views
From the discontinuity of $f(x)$ and $g(x)$, can we directly tell about the discontinuity of $f(g(x))$?
From the discontinuity of $f(x)$ and $g(x)$, can we directly tell about the discontinuity of $f(g(x))$?
I thought $f(g(x))$ would be discontinuous where $g(x)$ is discontinuous and where $g(x)=c$, ...
- 7,954
2
votes
0
answers
57
views
Discontinuous commuting function
Can two commuting (composition of the functions satisfies commutativity) with $f\ne g$ and both $f$,$g$ increasing functions on $[0,1]$ both be discontinuous on the set of rationals?
Context: I had ...
- 31
1
vote
0
answers
22
views
Cardinality of a collection of functions whose composition commutes
Consider a collection of functions $\mathcal F$ where (i) each individual element is a strictly increasing function from [0,1] to [0,1]; (ii) for any $f,g$ $f \le g$ or $f \ge g$ and $f\ne g$; and (...
- 31
1
vote
1
answer
26
views
Generalizing a proof about a preserved property under composition
There is a property of binary operations (functions from $\mathbb{S}^2$ to $\mathbb{S}$ for an arbirtary set $\mathbb{S}$) that I'm trying to figure out whether or not it is preserved. The cleanest ...
- 11
0
votes
0
answers
26
views
Time derivative a nonsmooth convex function. Chain rule.
Let us consider a convex function $f : \mathbb{R}^n \rightarrow \mathbb{R}$. Let us consider that is composed with an absolutely continuous function of a real variable $t$, $ x : \mathbb{R} \...
- 1
1
vote
0
answers
31
views
How to express the variance of a of a multivariate Gamma with a partial composition with an invGamma
I want to express the variance of a posterior. The posterior is expressed as a partial composition with a Gamma and inverse-Gamma:
$\Pi(\alpha,t)=Gamma(t,r(t,\alpha))$ with $r(t,\alpha)\sim invGamma(t,...
1
vote
0
answers
31
views
Is a norm of a vector of distances itself a metric on vectors?
Suppose I have $\vec x = (x_1, \ldots, x_n)$ and $\vec y = (y_1, \ldots, y_n)$ and a sequence of metrics $d_i$ for $i \in \{ 1, \ldots, n\}$ that are used for the $i$th component.
Consider a vector of ...
- 1,770
1
vote
2
answers
104
views
Show that if $g\circ f$ is injective, then $f$ must be injective. Is it true that $g$ must also be injective? [duplicate]
I am now self-studying Terence Tao's Analysis 1. I am trying to solve all of the exercises. The question I have a problem with is
Let $f: X\rightarrow Y$ and $g:Y\rightarrow Z$ be functions. Show ...
- 13
-1
votes
1
answer
51
views
How to intuitively think about composition of relations?
Examples:
$A \cap B$
elements that are in both $A$ and $B$
$A \cup B$
elements that are in either $A$ or $B$
$R \subset A\times B$
coordinates where the first coordinate is an element of $A$, and ...
6
votes
2
answers
152
views
Is $f(g(x))$ discontinuous?
Question:
Let $f(x) = \frac{1}{15x^2+8x+1} $ and $ g(x)= \frac{1}{(x-1)(x-2)} $, then the number of points of discontinuity of $f(g(x))$ is?
The answer key claimed that the answer is $1$, but I don't ...
- 783
0
votes
0
answers
21
views
Low rank function decomposition — necessary and sufficient conditions
Given a function ${\bf f} (x,y,z) = (f_1, \dots, f_n) : \mathbb{R}^3 \to \mathbb{R}^n$, suppose I can write ${\bf f} (x,y,z) = ({\bf g} \circ p)(x,y,z)$ where $p(x,y,z) : \mathbb{R}^3 \to \mathbb{R}$ ...
- 585
4
votes
1
answer
104
views
If $h:[0,1]\to [0,1]$ is continuous and surjective, then does there exist a continuous function $f$ such that $ff=h?$ If so, is $f$ unique?
Let $h:[0,1]\to [0,1]$ be a surjective continuous real function. There surely exist many functions $f:[0,1]\to [0,1]$ such that $f(f(x)) = h(x)\ $ on $x\in [0,1].$
Does there exist a continuous ...
- 17.8k
0
votes
0
answers
24
views
Vector-valued function composition
I was going through the blog article at Matrix Calculus. Under the section Vector chain rule they did some function composition thing to demonstrate the chain rule. The example that they took is as ...
- 125
0
votes
1
answer
60
views
Let the functions $f$, $g$ and $h$ be given, compute $(f \circ g \circ h)(4)$
I will appreciate so much someone helping me to verify this exercise.
Let the functions $f$, $g$ and $h$ be given by:
$f(x)=\begin{cases}
\dfrac{3+x}{x^2+1},&x\in~]-\infty, -2[\\
15+2x-x^2,&...
- 1,129
0
votes
0
answers
79
views
Under what conditions is $f = g^2$ convex?
Suppose that $x \in {\Bbb R}^n$ and $g: {\Bbb R}^n \to {\Bbb R}$. Under what conditions is $f := g^2$ convex?
It can be seen as the composition $f(x) = h(g(x))$ where $h(y) = y^2$.
From page 83 of ...
- 173
0
votes
0
answers
24
views
Functions that preserve the minimum inside an integral
Let $h(x, y): A \times B \rightarrow (0 , M]$ where $A = [0, a]$ and $B$ are two compact sets in $\mathbb{R}$ and M is a positive constant.
Write
$$f(y) = \int_0^a h(x,y) dx$$
I know that $y_0$ is an ...
- 193
3
votes
1
answer
93
views
Expected value of a random composition of functions
Let's say I have two functions, $f(x)$ and $g(x)$. For simplicity let's say both functions are linear.
Let $C_m(x)$ be any function where $f$ and $g$ are composed with each other exactly $m$ times (so ...
- 999
0
votes
2
answers
69
views
Limit of a composition function exists, but limit of the outer function does not exists
Limit of a composition function exists, but limit of the outer function does not exists
Given:
$f,g$ are real functions
$\lim_{x \rightarrow x_0}g(x)=u_0$
$\exists \delta > 0, \forall x \in \...
2
votes
0
answers
66
views
Let $f:[0,1]\to \mathbb{R},f(x)=4x(1-x),f_n(x)=f(f_{n-1}(x))\;\forall n\geq 1$ and $f_0(x)=x$. Find number of solutions to the equation $f_n(x)=x$.
Let $f:[0,1]\to \mathbb{R},f(x)=4x(1-x),f_n(x)=f(f_{n-1}(x))\;\forall n\geq 1$ and $f_0(x)=x$.
Find number of solutions to the equation $f_n(x)=x$.
My Attempt
I tried plotting graphs of $y=f(x)$ and $...
- 8,344
5
votes
1
answer
85
views
$f(n) = \frac{n^2 + n + 4}{2}, g(f(n)) = f(g(n))$ such that $g(n)$ is an integer.
Let $n$ be a strict positive integer.
Lets define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$f(1) = 3$$
$$f := {3,5,8,12,17,23,30,38,47,...}$$
$$f(17) = 155$$
etc
Notice
$$3+...
- 14.6k
6
votes
2
answers
228
views
What are the procedures to analyze the similarity of graphs?
This question is taken from a very challenging calculus problems book called the Advanced Problems in Mathematics by Vikas Gupta
And I have absolutely no clue on how to approach questions of the kind....
0
votes
2
answers
75
views
Prove that $\tanh (\cosh^{-1} (x)) = \frac{\sqrt{x^2-1}}{x}$
Prove that
$
\tanh (\cosh^{-1} (x)) = \dfrac{\sqrt{x^2-1}}{x}
$
,
by using that
$
\tanh (x) = \dfrac{e^x-e^{-x}}{e^x+e^{-x}}
$
,
and that
$
\cosh^{-1} (x) = \ln (x+\sqrt{x^2-1})
$
.
After substituting ...
- 13
0
votes
2
answers
146
views
Find $f^{(25)}(x)$ if $f(x)=x^{-3}$ by first finding general solution. [closed]
I don't get it what is the question asking for either composition or derivative of f 25th times.
Solution 1.
Sol:
$f(x)=x^{-3}=1/x^3$
$f^{(2)}(x)=f(f(x))=f(1/x^3)=1/(1/x^3)^3=1/(1/x^9)=x^9$
$f^{(3)}(x)...
1
vote
5
answers
85
views
Given function $f(x)$ is $1-\sqrt{4-2x}$ and $g(x)$ is $3-2x$, find a function $h(x)$ where $h(g(x)) = f(x)$
My steps:
$h(3-2x(x)) = 1-\sqrt{4-2x}$
$1-\sqrt{4-2(3-2x)}$
$1-\sqrt{4-6+4x}$
$1-\sqrt{-2+4x}$
But apparently, the correct answer is $1-\sqrt{1+x}$. Can you please explain?
- 53
3
votes
1
answer
120
views
How to cancel this Jacobi amplitude, elliptic integral, and incomplete beta function composition?
Problem and Context:
$\def\K{\operatorname K}\def\F{\operatorname F} \def\sn{\operatorname{sn}} \def\B{\operatorname B} \def\I{\operatorname I} \def\E{\operatorname E}\def\am{\operatorname{am}}$
An ...
- 10.1k
3
votes
2
answers
66
views
Compose a function with a parameter and given local maximum points.
I need to find a function $f_k: \mathbb R \rightarrow \mathbb R$ which includes the parameter $k \in \mathbb R$. The local maximum point of this function changes with the parameter $k$. This point is ...
- 135
0
votes
1
answer
58
views
composition relations in Set Theory
I have provided my solution and I see there are subtle differences to those provide in picture. Is my solution equivalent? Which is more rigorously correct?
\begin{align}
(S \circ R)^{-1} & = \...
- 235
1
vote
1
answer
44
views
If f: A $\rightarrow$ B, and f is onto and one-to-one, and we have functions g: B $\rightarrow$ A and h: B $\rightarrow$ A, show g=h.
The question highlights in addition that:
f $\circ$ g = f $\circ$ h = Id$_B$, and g$\circ$f = h $\circ$ f = Id$_A$
I understand that in order to prove g=h, you must show that
the codomain and domain ...
- 33
1
vote
1
answer
43
views
Centralizers of a function $f$, specify $g$ such that $f\circ g=g \circ f$
Let $f:R^n \to R^n$. I am looking for references on the description of centralizers of $f$, namely functions $g$ such that $f\circ g = g\circ f$.
Obviously $f^n$ (meaning the $n$-th iterate of $f$ ...
- 3,672
6
votes
1
answer
135
views
Do the lifts of commuting circle homeomorphisms commute?
I am going over proofs of the following result concerning the rotation number of orientation-preserving homeomorphisms of the circle: If $f, g \in \text{Homeo}^+(S^1)$ commute with respect to ...
- 1,153
6
votes
3
answers
257
views
If $\lim_{n \rightarrow \infty}\sum_{k=0}^np_{k}=\infty$ show that $f$ has a fixed point
We have $(p_{n})_{n\geq0}$ a sequence of strictly positive real numbers and $a$ a real number and the continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the sequence $$\left(\frac{\...
- 1,161
4
votes
1
answer
72
views
What is the neatest formula for the coefficients of the composition $f\circ g$ where $f,g$ are formal polynomials or generating functions?
There are well-known formulas for the coefficients resulting from multiplying two formal polynomials or generating functions (we define formal polynomial to be a generating function such that all ...
- 3,355
2
votes
1
answer
50
views
Solution to a differential equation of the form $f(x) - F(u(x)) = 0$ where $u(x) = x^n$ and $n \in \mathbb{N}$
While studying Maxwell's derivation for the distribution of molecule speeds in an ideal gas, I asked myself the following question: What would be the solution to a differential equation of the form: $$...
- 183
0
votes
0
answers
37
views
Matrix power verses iterated function
$ \text{If}\quad
A =\left [ \begin{matrix}
a & b \\
c & d \\
\end{matrix} \right ]\quad\text{and}\quad
A^n =\left [ \begin{matrix}
a_n & b_n \\
c_n & d_n \\
\...
- 2,120
6
votes
1
answer
101
views
Another strange limit which seems to converges to $\gamma$
Hi in trying to show that the Euler Mascheroni constant is irrational or not I find empiricaly :
$$\lim_{n\to\infty,x\to 0}f_n(x)=\lim_{n\to\infty,x\to 0}\frac{x!!!...!^{x!!...!^{x!...!^{...^{x!}}}}-x!...
- 3,705