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Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

3
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2answers
63 views

What is the $n$-time iterated adjugate of an $n\times n$ matrix $A$?

What is $\underbrace{\text{adj}\Big(\text{adj}\big(\ldots(\text{adj}}_{n\text{ adj}}\ A)\ldots\big)\Big)$, where $\text{adj}$ is written $n$ times, and the order of the matrix $A$ is $n\times n$? Can ...
7
votes
2answers
76 views

Find function $f(x)$ that satisfying differential relation

Suppose the functions $F(x)$ and $G(x)$ satisfying $$F(x)=f(x)-\frac{1}{f(x)}$$ $$G(x)=f(x)+\frac{1}{f(x)}$$ such that $F'(x)=(G\circ G)(x)$, with initial condition $f(\frac{\pi}{4})=1$ is given....
0
votes
0answers
15 views

Derivative of an infinite composition of functions

Let $f(x)=g(g(g(...g(x))))$, where the function $g$ is applied to $x$ and infinite amount of times. I am assuming that $x$ is real. What is special about points at which $df/dx=0$? Are they related to ...
3
votes
1answer
23 views

Limit of the composistion of two functions when the limit of $f$ does not exist?

The theorem about the limit of composition of two real functions $f$ and $g$ is proved here. But it is required that the two limits (of $f$ and $g$) both exist. I can't understand how to deal with ...
1
vote
3answers
34 views

If $(f\circ g)(x)=\tan^2x$ and $g(x)=\sqrt{\cos2x}$ then find $f(x)=?$

We've given : $$(f\circ g)(x)=\tan^2x$$ and $$g(x)=\sqrt{\cos 2x}$$ Then how to find the function $f(x)$? I know that $$(f\circ g)(x)=f(g(x))= f( \sqrt{\cos2x})$$ But I do not know how to find $f(x)$...
1
vote
1answer
24 views

Can composition of morphisms in a category be carried out on any subgraph of a commutative diagram, in-place?

Here is what the rule looks like to us and how we specify it to the app I'm writing. I was wondering can you take any commutative diagram $J$ and apply this rule to a subgraph matching $A \...
0
votes
0answers
19 views

What is the n-fold composition of a linear map “T”, if n is zero?

I am doing an assignment on Groups of Linear Transformations. Here is a defnition with regards to n-fold compositions: "Let $T \in \mathcal{L}(V)$ denote the n-fold compositions of $T$ with itself as ...
0
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0answers
28 views

How do I use the Composite of Continuous Functions theorem to show that a function is continuous?

$$f(x)= \sqrt \frac{x}{x+1}$$ I know that I have to split it into two separate functions: $$ g(x)= \sqrt x$$ $$h(x)=\frac{x}{x+1}$$ I'm just not sure what to do next, and I can't seem to find anything ...
0
votes
1answer
19 views

How do describe the composition of an indicator function?

I have an indicator function: I have to describe the composition in this form: I replaced the 'n' value with the function but I cannot simplify it, how can one describe and simplify the composition ...
0
votes
1answer
18 views

Proposition involving inner product and derivatives

I've found this proposition and I don't know how to prove it. The proposition is the following: Let's have $f:U\subset E\rightarrow R^{n}$, $E$ normed, $U$ open in $E$. Let's pick $p,q \in U, [p,q] \...
-1
votes
1answer
16 views

Is this an unsolvable function composition problem? Or an absurd?

I am solving a PDE problem. In a certain point, I achieve the following equality: $G(3x+1) = y^2 + e^x$ This is,basically, a problem involving function composition. How do I find $G(x)$? I have ...
0
votes
3answers
42 views

Given 2 functions $f,g:\mathbb{R} \to \mathbb{R}$ with $g \circ f$ surjective. Show that $f$ does not need to be surjective.

A friend has given me this task and asked for help. I immediately tried picking various $f$ and $g$ such that $g$ and $g\circ f$ are both surjective but $f$ is not. However, i always found myself ...
0
votes
1answer
29 views

Function composition according to relation

Let $f: A \rightarrow A \ $ and a relation $R_f$ over $A$ $\, $s.t.: $R_f = \left\{(x,y) \in A \times A \mid y=f(x)\right\}$ Show if the following statements are true or false and explain your choice: ...
0
votes
1answer
45 views

What is the domain and range of this relation in?

From: Triamudom Add.math sheet pg.19 This is the progress I’ve done so far. As you can see I couldn’t find a way to arrange $x$ in terms of $y $ or vice versa which is required to solve for the ...
4
votes
4answers
47 views

If limit of $f$ is $L$ and limit of $g$ is $M$, then limit of $g$ composed $f$ is $M$?

Problem: Find examples of functions $f$ and $g$ defined on $\mathbb{R}$ with $\lim\limits_{x\to a}f(x) = L$, $\lim\limits_{y\to L}g(y) = M$, and $\lim\limits_{x\to a} g(f(x))\neq M$. I have tried ...
2
votes
0answers
53 views

Solving equations with composed constrained functions

I was lately curious about an iterative approach that would solve maths equations containing composed functions with contraints. For example, if I have the following equation: $$ f(g(h(w))) = 0 \...
0
votes
3answers
13 views

Find the composition f(g(x)) when four functions are given

Determine $f \circ g$ for the following functions: $$ f(x) = \begin{cases} -x, & x < 0\\ x+1, & x \ge 0 \end{cases} \quad \text{and} \quad g(x) = \begin{cases} x^2, & x \le 2\\ x+2, &...
0
votes
2answers
17 views

How to find domain of complicated composite functions

If I was asked to find the domain of arccos($e^x$), are there universal steps I can take to be able to find the domain? I know that you want the inner function in f(g(x)) to be defined as well as the ...
0
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0answers
13 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), (...
18
votes
1answer
338 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
19 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
126 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
62 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
56 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
1answer
73 views

Proving f is onto given if $g \circ f = h \circ f$, then g = h

I realize that there has already been an answer to this problem. But I want to know if my proof was correct. Thank you for your time. Problem Suppose A,B, and C are sets and $f: A \rightarrow B$ ...
2
votes
2answers
42 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
32 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
33 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 ...
41
votes
2answers
620 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
58 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
30 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
81 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
42 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
19 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
79 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
89 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
76 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
414 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
31 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
199 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
108 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
89 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
44 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
44 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), &\;\;\;\;\;...