For questions about the composition of functions: If $f\colon A\to B$ and $g\colon B\to C$ are functions, their composition $g\circ f\colon A\to C$ is given by $x\mapsto g(f(x))$.

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Total Derivative and Composition

We are given that $C$ is a function of $Y_D$ and $Y_D=Y-Y\tau$. What would be the total differential of $Y=C(Y_D)$? So far I have the following: $$ dY=C_{Y_D}(1-\tau)dY+C_{Y_D}(-Y)d\tau$$ However I ...
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3answers
40 views

how to calculate derivative of $f_n(x)=f \circ f … \circ f(x)$? Derivative on $f \circ f_{n-1}$ or $f_{n-1} \circ f$?

Denote $f_n(x)=f \circ f ... \circ f(x)$, the $n$th power of composition multiplication of $f(x)$. Assume $f(x)$ is differentiable for any order. $f(1)=1$, $f^{'}(1)=p$, $f^{''}(1)=q$ Question: Get ...
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1answer
24 views

Which functions are the composition of convex functions?

The composition of convex functions is not necessarily convex or concave: For example, composing $f_1(x) = x^2-1$ and $f_2(y) = y^2$ gives $f_2(f_1(x)) = (x^2-1)^2$. Or consider $f_1(x) = x^2$ and ...
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1answer
19 views

Discrete Math Compositions

I am having trouble with these compositions. $$T = \{(a,a), (a,b), (b,c), (b,d), (c,d), (d,a), (d,b)\}$$ $$U = \{(a,a), (a,d), (b,c), (b,d), (c,a), (d,d)\}$$ I need to find $T \circ T$, $U \circ T$, ...
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0answers
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Existence of a limit - Composition of continuous functions - Questioning [duplicate]

The question of Jim Darson to this link, Don Antonio replied using a similar property in the composite of continuous functions ($\frac{\text{Re}\,z}z$ and the line $\;y=mx\;$) is continuous, but with ...
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1answer
54 views

In which cases are $(f\circ g)(x) = (g\circ f)(x)$?

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.
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1answer
33 views

Inverse of the composition of two functions

If I have a composition of two functions: $$y = f(g(x),h(x))$$ where both $g(x)$ and $h(x)$ are readily invertible, can I find the inverse of the composition? i.e.: Can I find $x = f^{-1}(y)$? I ...
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1answer
22 views

What is the composition of the two given relations $R_1\circ R_2$?

I have a set $A = \{a,b,c,d\}$ on which two relations are $R_1=\{(a,b),(a,d),(b,c),(c,a),(c,d),(d,b)\}$ and $R_2=\{(a,b),(b,c),(d,c),(a,d),(a,c)\}$. What will $R_1\circ R_2$ be? $\circ$ is a the ...
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12 views

Composition of continuous fonctions VS Limit of composite functions

We know that the composition of continuous fonctions is continuous, but we don't have an analogous property for the limit of composite functions. Is there anyone that could explicitly explain why this ...
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1answer
24 views

Generating Infinite Set with Function Composition

I imagined myself today being infinitely small, standing on the inside of a closed and perfectly mirrored surface and holding a laser. Could this surface be shaped in some way where I could turn on ...
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0answers
15 views

Finite Permutation Composition

This is a problem I'm trying to solve. Given a permutation $ \sigma $ on a finite set $ \mathcal{A} $ of order $ n $, show that there exists a positive integer $ 0< k \leq n $ such that $$\bigl( ...
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2answers
15 views

How to draw a diagram defining a map as the composition of two other maps?

I would like to know what is the most common way of drawing a diagram to define a certain map $h: X \to Z$ as the composition of some two mappings: $f : X \to Y$ and $g: Y \to Z$.
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1answer
54 views

Prove $(g+h)\circ f=g\circ f+ h\circ f$ [closed]

Let $g,h,f$ be functions with domains and ranges on the real numbers. I have to prove that $$(g+h)\circ f=g\circ f + h\circ f$$ It seems so simple, but I don't know where to start the proof. Maybe ...
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27 views

Composition of $C^ i$ functions

It is clear that the composition of two $C^i$ function is still a $C^i$. But my question is more about a kind of its reciprocal. Let's consider two open interval $I_1$ and $I_2$ in $\mathbb{R}$. ...
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1answer
10 views

Substituting functions into other functions in computability, need help with Cutland

I'm working my way through the Cutland text on computability and I'm having a little trouble understanding exactly what he's saying in regards to substituting functions into other functions (if you ...
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4answers
69 views

$g\circ f$ bijective iff $f$ and $g$ bijective? [duplicate]

Is the following true: $g\circ f$ bijective iff $f$ and $g$ bijective? Or can the requirements be weakened for $g$ (i.e. $g$ only injective or surjective)? Or $f$?
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2answers
48 views

lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$ in Metric Space - Implications

A Metric Space $<M,d>$ is given by the Metric $M$ and distance function $d$ If there exists a Cauchy Sequence $x_n$ such that: lim$_{n→∞}d(x_n, a) \neq d($lim$_{n→∞}x_n, a)$, for some $a \in ...
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2answers
54 views

O(1) solution for number of times to apply iterated function

Is there an O(1) solution for finding the number of times to apply an simple iterated function to satisfy an inequality? For example, if the function is $$f(n) = 0.5n - 10; n > 100$$ and we want ...
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1answer
25 views

Four Problems Concerning Orbits of Biconditional Functions $f: A \rightarrow A$

Here is a problem. There are four proofs that I attempted to do but am not sure are correct. Any insight or thoughts concerning the problem and my solutions are most appreciated. Let $A$ be ...
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1answer
32 views

Basic of $n$-fold iterations

Here is an excerpt of a problem: Let $A$ be nonempty set, and let $f:A \rightarrow A$ be a function. Suppose that $f$ is bijective. For each $n \in \mathbb{N}$, let $f^n$ denote the function $A ...
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0answers
48 views

Proving an identity of the composition of the delta distribution with a differentiable function

Given a differentiable function $f$, some $x_j$ ($j \in \{1, ..., n\}$) such that $f(x_j) = 0$ $\forall j$ and $f'(x_j) \ne 0$ $\forall j$, and the following definition of the composition of a ...
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1answer
22 views

Finding univalent function with $f(z_1)=f(z_2)$

Let $\Omega$ be a simply-connected domain. Let $z_1,z_2\in\Omega$. Prove that exists an univalent function such that $f(\Omega)=\Omega$ and $f(z_1)=z_2$. Since $\Omega$ is simply connected, one ...
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2answers
29 views

Basic algebra (composite functions)

The question is $h(x)=\frac{(1+x)}{(1-x)}$ find $h(1-x)$ I understand how to solve the question, it's: $1+\frac{(1+x)}{(1-x)}$ What I can't seem to understand is why the denominator $1-(1-x)$ ...
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1answer
29 views

Domain of piecewise-defined functions composition

I'm wondering what is the right way to perform function composition on those two functions: $$f\left(x\right) = \left\{ \begin{array}{lr} 1/x & : x \ne 0\\ 0 & : x = 0 ...
3
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1answer
36 views

Is it possible to find $g^k(x)$?

Given $g(x)=\frac{x}{2}+\frac{1}{x}$, is it possible to find an expression for $g^k(x)=(g\underbrace{\circ \cdots \circ}_k g)(x)$, where $k$ is some positive whole number? For example, given ...
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1answer
46 views

Find $f \circ g$, $g \circ f$, $f \circ f$, $g \circ g$ where $f(x)=-1+|x-2|$ and $g(x)=2-|x|$.

If $f(x)=-1+|x-2|;0\leq x\leq4$ and $g(x)=2-|x|;-1\leq x\leq3$ Then find $f \circ g(x),g \circ f(x),f \circ f(x),g \circ g(x)$ In the above figure the red graph shows $g(x)$ and the blue graph ...
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3answers
35 views

Let $x^2+kx=0;k$ is a real number .The set of values of $k$ for which the equation $f(x)=0$ and $f(f(x))=0$ have same real solution set.

Let $f(x)=x^2+kx;k$ is a real number.The set of values of $k$ for which the equation $f(x)=0$ and $f(f(x))=0$ have same real solution set. The equation $x^2+kx=0$ has solutions $x=0,-k$. So the ...
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2answers
48 views

Can any function be written as a composition of other functions?

Can any function be written as a composition of other functions? For example, can a polynomial $h(x)$ be written as $k(g(x))$?
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1answer
99 views

Can it be that $f$ and $g$ are everywhere continuous but nowhere differentiable but that $f \circ g$ is differentiable?

So, I was just asking myself can something like this happen? I was thinking about some everywhere continuous but nowhere differentiable functions $f$ and $g$ and the natural question arose on can the ...
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2answers
22 views

Composition of sets

Question: Let the set $A$ be defined as $A = \{ a, b, c, d \}$, and let the relations $R$ and $S$ on the set $A$ be defined as $R = \{(d, a), (a, b), (b, c)\}$, and $S = \{(a, a), (b, d), (d, c)\}$. ...
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2answers
40 views

Find the number of real solutions of the equation $f(f(x))$=4

Consider the graph of a real-valued continuous function $f(x)$ defined on $R$(the set of all real numbers) as shown below: Find the number of real solutions of the equation $f(f(x))=4.$ I found ...
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0answers
16 views

Composition of various combinations of one one,many one,onto and into functions.

Let $f:R\to R$ be an one-one function,$g:R\to R$ be a many one function,$h:R\to R$ be an onto function and $l:R\to R$ be an into function. $(1)$I know that one one function composition one one ...
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1answer
20 views

If $f:R\to R$ and $g:R\to R$ be functions such that $g(x)$ is onto and $fog(x)$ is injective then prove that $g$ must be injective.

If $f:R\to R$ and $g:R\to R$ be functions such that $g(x)$ is onto and $fog(x)$ is injective then prove that $g$ must be injective. I dont know how to prove it.I only know that composition of two ...
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2answers
18 views

Proving the chain rule of a given function

Suppose that $f'(2)=3$, $f'(5)=4$, and let $h(x)$ be the composite function $h(x) = f(x^2+1)$. Find $h'(2)$ I get how to prove the $f'g(x)*g'(x)$ part, which leads to $4*g'(2)$ but how do I prove ...
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1answer
42 views

How do I find $f(0)$ for this function?

Let $f: \mathbb R \to \mathbb R $ such that $$(f \circ f \circ f)(x)= (f\circ f)(x)+x$$ for every $x \in \mathbb R$. How can I compute $f(0)$?
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2answers
64 views

Find all triples (a,b,c) such that h(h(x))=x, and a,b and c are non-zero real numbers

Suppose that $a,b$ and $c$ are non-zero real numbers. Define $$h(x) = \frac{ax+b}{bx+c}$$ for $x\neq -\frac cb$. Determine all triples $(a,b,c)$ for which $h(h(x)) =x$ for every real number $x\neq ...
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1answer
94 views

$f^m = f^k$ Proofs

$1$. Prove that for any $f :J_n → J_n $, there exists a positive integer $m$ such that $f^m = f^k$ for some positive integer $k < m$. $2.$ Let $f :J_n → J_n$ be a function, and let $m$ and $k$ be ...
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3answers
113 views

Why does Arccos(Sin(x)) look like this??

I can kind of understand the main direction (slope) of $y$ over the different $x$ intervals, but I can't figure out why the values of $y$ take on the shape of straight lines and not curves looking ...
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1answer
49 views

name or characterization for the “partition” lattice of integer partitions of some n? (Young lattice row partial order)

Is there a name or characterization for the "partition" lattice of integer partitions of some n? Young's Lattice depicts the integer partitions of numbers. Often Young diagrams are used in displaying ...
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1answer
147 views

How to simplify this composition function $g(x)=\underset{n\text{ times}}{\underbrace{f \circ f \circ f \circ f\circ f \circ \cdots\circ f}}(x)$?

Let $f(x)=\dfrac{\sin x}{(1+\sin^n(x))^{1/n}}$ and $g(x)=\underset{n\text{ times}}{\underbrace{f \circ f \circ f \circ f\circ f \circ \cdots\circ f}}(x)$. Where $\circ$ represents function ...
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4answers
46 views

Understanding Theorems on Composition of functions

Consider below two theorems about Injective and Surjective mappings Theorem 1: If $f: A ...
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0answers
71 views

Computing the $n_{th}$ Derivative of $f(x)^m$

I know that there is a general formula to compute the $n_{th}$ derivative of a composition as it is mentioned in this post. The formula is named Faà di Bruno's formula. However, it is really ugly and ...
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2answers
25 views

Associativity of function composition?

I am working on an exercises that should show that automorphism under composition satisfy a group definition, there are basically four things I need to prove: 1.closure, 2.associativity, ...
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2answers
24 views

Discrete and combinatoric mathematics (Functions)

$f = ax^2 - b$ and $g = cx + d$ Where $a,b,c,d$ are all coefficients. Find $a,b,c,d$ when $f◦g = g◦f$. Here is what I have: \begin{align*} f◦g &= a(cx + d)^2 -b = a(c^2)(x^2) + ...
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1answer
50 views

limit of composite functions- switching order of limit

When is it correct to do stuff like this: $$\lim\limits_{x\to a}f(g(x))=f(\lim\limits_{x\to a}g(x)) \quad (*)$$ I know (*) is true when $g$ is continuous at $a$ and $f$ is continuous at $g(a)$. ...
2
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2answers
50 views

Supermum of $f(x)=\frac{x}{\sqrt{1+x^2}}$ composed $n$ times on itself [closed]

For a natural $n$ and $f(x)=\frac{x}{\sqrt{1+x^2}}$, find the supremum of the function: $f^n(x)=f\circ f\circ...\circ f(x)$, when $f$ is composed $n$ times on itself.
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1answer
17 views

half composite function

There is half derivative. Is there any definition of half composite function? Or is it possible to define half composite of a function? 1) $f^n=f \circ \cdots \circ f $ ($n$-times, $n \in \mathbb ...
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2answers
66 views

How to find the composition of case-defined functions?

Let $$g(x)= \begin{cases} 3+x & \text{if $x\leq0$} \\ 3-x &\text{if $x > 0$} \end{cases}$$ Find $f$ if $f$ is defined as $f(x) = g(g(x))$. How to solve the problem ...
1
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1answer
29 views

A quadratic polynomial $f$ such that $f\circ f' = f'\circ f$

Given that $\ f\left( x \right)=ax^{2}+bx+c$, find a value for each of $a, b$ and $c$ such that: $f\left( f'\left( x \right) \right)=f'\left( f\left( x \right) \right)$. What I did: ...
2
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0answers
56 views

Composite function between manifolds

I have this claim left as an exercise in my course: Let $f:M\to N$ be some function between two smooth manifolds $M$ and $N$ (respectively of dimensions $m$ and $n$). Prove that, if for any smooth ...