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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How to solve diff. eq. involving function taking expression including itself as variable

$$ f(x)=f \left( x \pm \frac{l}{\sqrt{1+\dot{f}^2}} \right) \mp \frac{l}{\sqrt{1+\dot{f}^2}}\dot{f}^2, $$ where $l$ is a constant. How is such a beast even approached? If anyone got intuition for ...
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1answer
50 views

Suppose $f(x)$ and $g(x)$ are $1$−$1$ functions on their respective domains. Show that $f(g(x))$ is a $1$−$1$ function.

I have an idea of where to go with this proof, but would like a second opinion as to wether I have actually made a logical argument. $f(g(x)) \neq f(g(y))$ where $x, y \in \mathbb{R}$ and $x \neq y$. ...
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2answers
57 views

Composing Even and Odd Functions With Floors

I have a question set that is asking me to determine if the following two statements are true and justify my answer. If $f(x)$ is even then $g(x)=[f(x)] $is even and If$ f(x)$ is odd then$ ...
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34 views

Let $h(z) = g(f(z))$. If two of the three functions $f$, $g$, and $h$ are holomorphic and non-constant, must the third also be holomorphic?

If $h$ and $g$ are holomorphic it seems like the answer is no. Let $f(z) = f(re^{i\theta}) = \sqrt re^{i\theta/2}$ for $\theta \in [0,2\pi)$, and let $g(z)=z^2$. Then $f$ is discontinuous on the ...
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0answers
17 views

Finding the $h'(x,y,z)$ if $h= p \circ q $ $p(x,y,z)=(x \sin y, x \cos y, z+y ), q(x,y,z)=(x^2,x+y,2e^z)$

I just want someone to check my work basically. Providing thoughts and insight, into possible mistakes: Finding the $$h'(x,y,z)$$ if $$h= p \circ q ,\ \ p(x,y,z)=(x \sin y, x \cos y, z+y ), \ \ ...
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2answers
34 views

Proof of onto and one-to-one functions, composition

I want to prove this: Let $f: A \to B$ and $g: B \to C $ be functions. if $g \circ f$ is onto, and $g$ is one-to-one, then f is onto. Here is what I have done, can someone please verify my work: ...
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1answer
30 views

Continuity of the composition of continuous functions

I have a question about the continuity of composite functions. Let $f:[a,b] \rightarrow \mathbb{R}$ and $g:[c,d] \rightarrow [0,1]$ be continuous functions. Define the function $h:[a,b]\times [c,d] ...
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4answers
94 views

Is is true that: $f \circ g = g \circ f \implies$ $f$ is linear or $g$ is linear?

Is the following statement true? In case not, what's a counterexample? Thank you. If $f,g: \mathbb R \to \mathbb R$ are two continuous functions satisfying $f \circ g = g \circ f$, then either $f$ ...
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0answers
26 views

PDE system for two composite functions

Can someone help? I have a PDE system for two unknown functions, $f(x,y,t)$ and $g(x,y,t)$ $$a_1(t)\frac{\partial f(x,y,t)}{\partial x} + a_2(t)\frac{\partial f(x,y,t)}{\partial y} + ...
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4answers
112 views

Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{(\frac{\pi}{\alpha^5})}^\frac{1}{2}$ [duplicate]

In my physics course this standard formula is used a lot without proof so it would be interesting to see a neat proof for it. From a previous thread by me I know the proof for $\int ...
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1answer
25 views

Derivatives of Implicit Functions (Abstract Case)

I have never been good at differentiation of implicit functions in cases when in a function is given, much less in abstract cases with composite functions. Hopefully someone can help me get started on ...
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2answers
32 views

Why is this proof for an arbitrary function constrained to a constant one?

Sorry if this seems trivial, I'm having some difficulty understanding a proof. I'm doing exercise 5.1.14 of Velleman's How to Prove It and a solution posted in this question, including the comments, ...
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1answer
45 views

Compose $(1243)$ and $(5)$

Checking my work. In either direction: $(1243)[1] = 2$ and $(5)[2] = 2$, so far we have $(1, 2,\ldots$ $(1243)[2] = 4$ and $(5)[4] = 4$, so far we have $(1, 2, 4,\ldots$ $(1243)[4] = 3$ and ...
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0answers
53 views

Composition factoring into nonnegative integer polynomials

Consider the integer polynomials with nonnegative coefficients, such as: $$ 1 + 2x +2x^2 $$ $$ 3 + 3x^4 + 11x^{10}$$ $$ 13 + 7x + x^2 $$ I asm interested in knowing "what is a composition ...
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1answer
59 views

General closed form solution to $f'(x) = P(f(x))/P(x)$

Does there exist a general closed form solution (in terms of elementary or special functions) to the differential equation: $$ \frac{df(x)}{dx} = \frac{P(f(x))}{P(x)} $$ when $P(x)$ is a polynomial ...
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1answer
38 views

Composition of analytic functions is analytic

I want to find a proof that shows the composition of two analytic functions is analytic. I know I should prove this using Cauchy-Riemann equations, but I wasn't able to use them in the proper way in ...
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0answers
16 views

Matrix representation of function concatenation using other basis than polynomials.

I have now familiarized myself with the Carleman-matrices which represent function composition of polynomials (actually taylor series terms) and built some of my own. I noticed that for any finite ...
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1answer
26 views

Function composition as representable by matrices?

I know from linear algebra that for different sets of functions differentiation can be expressed using matrix multiplication on a vector representation of the function. For instance polynomials and ...
2
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2answers
58 views

The limit of a composed function

Let $g_n: \mathbb{N} \rightarrow \mathbb{R}$ and $f_n(x): \mathbb{N\times R} \rightarrow \mathbb{R}$. If $g_n \rightarrow g$ and $f_n(g) \rightarrow f(g)$, can we deduce $f_n(g_n) \rightarrow f(g)$? ...
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3answers
65 views

Composite Function Equality

Let f, g be functions$: \mathbb{R} \mapsto \mathbb{R}$ with $f(x)=x^2+ax+b$ with $a,b \in \mathbb{R}$, such that $(f\circ g)=(g\circ f)$. If the equation $g(x)=x$ has precisely one solution for all ...
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1answer
53 views

Composition operators on fractional-order Sobolev spaces

Preliminaries: We know that the fractional-order Sobolev spaces $\mathrm{H}^s(\mathbb{R})$ and $\mathrm{H}^s(\mathbb{T})$ are closed under multiplication provided $s > 1/2$. This is proved for ...
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1answer
45 views

Find: $\lim_{x\to0}\left(\lim_{n\to\infty}2^{2n}\left(1-\left(f ^{\circ n}(x)\right)\right)\right)$

Let $$f:[0,1]\to\Bbb R\;\;\mbox{defined by}\;\;\; f(x)=\sqrt{\frac{1+x}{2}}$$ Find: $$\lim_{x\to0}\left(\lim_{n\to\infty}2^{2n}\left(1-\left(\overbrace {f \circ f \circ f \cdot\cdot\cdot \circ ...
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7answers
368 views

Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) ...
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0answers
19 views

Are these two compositions of two functions differentiable?

Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$. $f$ is Fréchet-differentiable in $0$, and ...
2
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1answer
29 views

Finding a (nonidentity) rational map of the plane with period $7$

Does there exist a nonidentity (which also is not a rotation) rational map $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ with period $7$, i.e., for which the seventh iteration $f^7$ is the identity ...
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0answers
16 views

Composition of Lp convergent function and continuous function

Let $f$ be a continuous function on $\mathbb R$ such that for $U\subset \mathbb R ^n$ bounded it holds that $\forall w\in L^p(U) ~~ f(w)\in L^q(U)$. Let $~u_k \rightarrow u$ in $L^p(U)$ . Does ...
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0answers
25 views

Convergence in probability of function composition.

I need to show that $G_n \stackrel{P}{\to}_n F_0$, i.e. for any $\epsilon>0$ $$ P(|| G_n - F_0||>\epsilon) \to_n 0 $$ We know the following: $G_n$ and $F_0$ are a bilinear functions from ...
1
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1answer
19 views

While finding points of discontinuity for a composite function do I need to consider the points of discontinuity of individual functions too.

I'm solving problems based on composition of functions and stuck in this problem. If $f(x)=\frac{1}{(x-1)(x-2)}$ and $g(x)=\frac{1}{x^2}$, then find the points of discontinuity of $f(g(x))$. We ...
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1answer
26 views

Struggling with a problem in functions.

Suppose '$f$' is a continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $f(f(a))=a$ for some $a \in \mathbb{R}$ then find the number of solutions of the equation $f(x)=x$. Options given: ...
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0answers
10 views

Composition of functions class $C^n$

Suppose I have 2 functions class $C^n$ and I consider their composition. Would that still be a $C^n$ function? If so why (I demand a proof). It seems logical to me that it is true but I can't find a ...
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3answers
79 views

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective.

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective. My attempt: Using contrapositive, if Y is not injective. then Y ∘ Y is not injective, the there exist x, x' ∈ X with x ≠ x' but ...
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1answer
37 views

Composing Linear Transformations

Hello and thank you in advance; The problem: "Let V be a vector space and T a linear operator $T:V\rightarrow V $, show that $$[T^m]_B =[T]_B^m$$ Where $B$ is a basis(any) of $V$ and $T^m=T\circ T ...
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1answer
51 views

which functions can be obtained as a composition of a continuous function with itself? [duplicate]

let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for ...
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1answer
28 views

How to find $g(x)$, if: $f(x)=(x+1)/x$, and $f(g(x))=x$?

How to find $g(x)$, if: $f(x)=\frac{(x+1)}{x}$, and $f(g(x))=x$? I know that the answer is that $g(x)=\frac{1}{(x-1)}$ But how to come to that answer remains a mystery to me Please give me some ...
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0answers
26 views

Differentiate $g\circ f$ transformation

Differentiate $g \circ f$ of the following functions: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$f(x,y)=(x-y,x+y)$$ $$g: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$g(x_1,x_2)=(e^{x_1} \cos ...
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1answer
16 views

How to determine a function from a sequence of consecutively composed functions?

Let $ f(x) = x+1 $ and $g(x) = 2x$ Prove $f^2g = gf $ and determine $f^igf^jgf^k(x)$ explicitly as a function of x and in terms of i,j,k. I got through the proof but I don't understand what the ...
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1answer
35 views

If a composition of two maps is smooth, as well as one of the maps, then so is the other.

Let $M$, $N$, and $K$ be smooth manifolds, and consider the maps $g:M\to N$, and $f:N\to L$. Assume that the composition $f\circ g$ is smooth. If any of $f$ and $g$ is smooth can we conclude that the ...
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1answer
32 views

Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
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2answers
186 views

The composition of a nowhere-differentiable function with a differentiable function.

This is actually Problem $ 17 $ from Chapter $ 10 $ of the Fourth Edition of Michael Spivak’s Calculus. The statement is quite simple, but I have not had any success in finding an example. Here is the ...
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3answers
92 views

If $ f $ is injective and $ g $ is injective, then $ f \circ g $ is surjective. [duplicate]

I can prove that if $ f $ and $ g $ are both injective, then $ f \circ g $ is injective, but I don’t know how to prove that $ f \circ g $ is surjective.
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2answers
49 views

Show that $g$ is one-one if and only if $g$ is onto.

Original problem A function $g$ from a set $X$ to itself satisfies $g^m=g^n$ for positive $m$ and $n$ with $m>n$. Here $g^n$ stands for $g\circ g\circ \dots g$(n times). Show that $g$ is one-one ...
0
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1answer
19 views

Taylor Expansion of Composition of Functions

I have in my lecture notes that $$f(g(x+h))-f(g(x))\approx f'(g(x))\left(g(x+h)-g(x)\right)+f''(g(x))(g(x+h)-g(x))^2$$ He explained can found via taylor expansion, but I try to expand it and am not ...
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0answers
45 views

Linear transformation of eigenspace is subset of eigenspace

Let $V$ be a vector space over a field $\mathbb{F}$ and let $L$, $M$ be two linear transformations from $V$ to itself. a. Show that the subset $W= {x ∈ V : L(x) = M(x)}$ is a subspace of $V$ b. ...
1
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1answer
42 views

Composition of 2 monotonic functions

Let $f$ be a monotonic function $f:[a,b] \rightarrow\mathbb{R}$ and $g$ be a monotonic function $g:[c,d]\rightarrow[a,b]$. Show that $f\circ g$ is monotonic
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2answers
20 views

Given two piece-wise functions, prove their composition is periodic.

I know how to graph h(f(x)), however, I am having trouble proving it is periodic. i. We must prove that h(f(x + k)) = h(f(x)) for all x ∈ ℝ and for some k ≠0 ∈ ℝ in order to prove that h(f(x)) is ...
0
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2answers
30 views

$f \circ f^{-1} = i_B$ proof using the fact that $f^{-1} \circ f = i_A$

Suppose f is function from A to B, and suppose that $f^{-1}$ is a function from B to A. Assume $f^{-1} \circ f = i_A$. Then show therefore that $f \circ f^{-1} = i_B$. I tried applying left ...
2
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0answers
78 views

We all know about compositions of functions, but what about decomposition. Is there a way with math, not just heuristics?

The composition operator is a well know and quite often used method in integration and differentiation, think u-substitution. However, given a composition like $$f(f(f(...f(x)...)))$$ Where there are ...
2
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0answers
31 views

functions compositions, three sets, counting compositions

Given three sets $P, Q, R$ such that $|P|=p, |Q|=q, |R|=r,$ and $p,q,r > 1$ let $f(x): P\rightarrow Q$, and $g(x):Q\rightarrow R$ be two functions. Find the number of functions which can be ...
1
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0answers
19 views

Internalizing results about composition and surjectivity/injectivity

I'm trying to see if there is any intuition pump / analogy that allows me to internalize ( and readily derive them) a series of results about the concepts of composition mixed with ...
2
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1answer
58 views

How to find $s(\exp(d(x)))$ ~ $ x + 2 $?

Let $x$ be a positive real. I want to find a pair of analytic functions $s(x),d(x)$ such that $s(d(x)) = x$ and $ s(\exp(d(x)))$ ~ $ x + 2 $ More presicely I Also want : $$ \lim_{x \to \infty} ...