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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Prove that if B(0) = 0, then A(B(x)) is a formal power series

I'm working through my Combinatorics textbook and am stuck on this proof. The textbook explains it pretty well, but I am having trouble with one of the steps. I was hoping I could get some help here ...
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1answer
34 views

Prove any function can be written as a composition between an injective and a surjective function.

Given an arbitrary function $f:A\rightarrow B$, write it as a composition between an injective and a surjective function, respectively.
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1answer
18 views

Question about the composite of a homeomorphism and a continuous onto function.

If $f : (G,T)$ homeomorphically to $(A,T_1)$, and $h: (A,T_1)$ continuously and onto $(C,T_2)$, then is it always the case that, given the composition $g = h \circ f : (G,T) \rightarrow (C,T_3)$, the ...
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1answer
70 views

What's the order of operations when dealing with function composition?

Given $f:[0,1]\rightarrow \mathbb{R}$ and $g:[0,1]\rightarrow [0,1]$, $g(x)=x^2$. Which of the two equalities is true? 1)$f^2(x^2)=f^2(g(x))=(f^2\circ g)(x)$; 2)$f^2(x^2)=f(x^2)\cdot f(x^2)=f(g(x))...
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Finding the upper derivitive of the compostion of two functions

Let $f$ be defined on $[a,b]$ and g a continuous function defined on $[\alpha , \beta ]$ that is differentiable at $\gamma \in (\alpha, \beta)$ with $g(\gamma)=c\in(a,b)$. Show that if $g'(\gamma)>...
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0answers
41 views

Circles and generic implicit functions

I have some problems understanding circles. $x^2+y^2 = 1$ is a circle. It defines equivalence class where all (x,y) points belonging to the circle are in the same equivalence class. $(\cos a, \sin a)$...
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2answers
46 views

Is $ f \circ g $ invertible in the diagram below?

I was working through Can the composition of two non-invertible functions be invertible? For the image below is $f \circ g$ invertible? Thanks!
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1answer
32 views

Precalculus questions: Domain, range, and composition of functions

Directions: evaluate each of the functions at the indicated value of $x$. construct each of the functions, then find the domain. If $f(x)=\{(3,5),(2,4),(1,7)\}$, $g(x)=\sqrt{x-3}$, $h(x)=\{(3,2),(4,3)...
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33 views

How to put derivative of composition in Jacobian matrix?

Here are two functions: $f\left(u,v\right)=u^{2}+3v^{2}$ $g\left(x,y\right)=\begin{pmatrix} e^{x}\cos y \\ e^{x}\sin y \end{pmatrix} $ I need to make Jacobian matrix of $f\circ g$. I found ...
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1answer
20 views

2 ways of solving derivative of composition of functions?

Functions: $f\left(u,v\right)=u^{2}+3v^{2}$ $c\left(t\right)=\begin{pmatrix} e^{t} \\ e^{-t} \end{pmatrix} $ I calculate composition and drivative on 2 ways: 1. substitution and 2. chain rule. ...
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1answer
48 views

When can composition of functions break?

Are there circumstances which may arise within, say, functional analysis where one's able to evaluate a composition of functions by only first evaluating the functions under composition obtaining real ...
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0answers
30 views

Prove $r \circ f = s \circ f \implies r =s$ holds for a surjection $f$.

Frankly I'm not convinced what I have is technically correct so I was hoping to get some verification. Suppose we have functions $f:A\rightarrow B$, $r:B\rightarrow C$, and $s:B\rightarrow C$. ...
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3answers
61 views

Disprove: $f\circ g = f \circ h \implies g=h$ for a surjective function $f$

I tried using a very specific counterexample here where I select a surjective function for which the compositions are equal but the functions within are not. This is probably off-base, but it's what ...
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2answers
55 views

Let $(f \circ g)(x) = x^2 +2x -1$. Find $f(x)$ if $g(x) = x+2$

Problem : Let $(f \circ g)(x) = x^2 +2x -1$. Find $f(x)$ if $g(x) = x+2$ My Attempted Solution $$f(g(x)) = x^2 +2x-1$$ $$f(x+2) = x^2 +2x -1$$ But that is as far as I got. The problem I'm having ...
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5answers
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For some arbitrarily fast growing function $f$ and a strictly sublinear function $g$, can $g \circ \cdots \circ g \circ f$ always grow polynomially?

Given two functions $f, g: \mathbb{R}_{≥ 0} \to \mathbb{R}_{≥ 0}$ that are monotonically growing, with $g(x) \in o(x)$ (i.e. $g$ grows strictly sublinear), does there always exist an $m \in \mathbb{N}$...
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0answers
129 views

What is the range of this function?

What is the range of $h$? $f(x)=4x+1$ $g(x)=(x-1)/3$ Let $h=\{f^n(g^m(1)):n,m\in\mathbb{N}\geq0\}$ What is the range of $h$? Show that $(2\mathbb{N}-1)\subset H$. ... okay I've done a bit more: ...
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0answers
20 views

Slicing a 3d surface using a 2d line equation

So what I'm trying to do is to find the equation of a 2d function on a 3d surface using a 2d line equation. With : $z = f(x, y)$ the equation of the surface and $ax + by + c = 0$ the line ...
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4answers
81 views

How to find the domain of this function?

f(x)= $\frac{(\sqrt{x}-\sqrt{x-1} )}{( \sqrt{x}+\sqrt{x-1} )}\;$ first off $\sqrt{x}$ is defined for: $$x > 0 \tag{1}$$ and $\sqrt{x-1}$ is defined for: $$x \ge 1 ...
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1answer
19 views

Defining composite of piecewise function

Let $$f(x) = \begin{cases} x+2, -3\leq x \lt-1 \\ x-1,-1\leq x \lt3 \end{cases}$$ I had to find $f(f(x)$ I defined $f(f(x)$ to be: $$f(f(x)) = \begin{cases} f(x)+2, -3\leq f(x) \lt-1 \\ f(x)-1,-1\...
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2answers
95 views

Find $f(x)$ if $f(f(x)) = x^2 -1$ [closed]

If $f(f(x)) = x^2-1$ Find $f(x)$ If there are more than one solutions find the family of functions that satisfies this.
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55 views

Flip/flop of finite joins and finite meets of lattices

Let $\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{C}$ be lattices (in fact in the example I have in mind, they are distributive and even co-Heyting lattices). Let maps $f:\mathfrak{A}\rightarrow\mathfrak{...
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0answers
36 views

Find all $g(x)$ such that $f(g(x))=g(f(x))$

For a given function $f(x)$ I want to find all functions $g(x)$ such that $f(g(x))=g(f(x))$. Two solutions are always $g(x)=x$ and $g(x)=f(x)$, but are there any more? Initially I wondered this for $...
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1answer
25 views

Function composition by Chain rule

I need to calculate by chain rule the partial derivative of: dz/du dz/dv dz/dw at point (u,v,w)=(2,1,0) for z=x^2+xy^3, x=uv^2+w^3, y=u+ve^w I don't know how to find the function composition.
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1answer
20 views

Composition of functions Discrete Math question

How do I do this? All help is appreciated! Would prefer a step by step tutorial but any help is ok :) Let $h= g\circ f\circ g$ where $f \colon \mathbb R \to \mathbb Z$ is the floor function and $...
2
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1answer
44 views

Functions: Relations

I'm stuck on this question and would appreciate the solution, thanks! Let $R$ and $S$ be the relations on $A = \{1,2,3,4,5\}$ given by$$R = \{(1,2),(2,3),(3,4),(4,5)\}\\S=\{(2,3),(2,4),(3,4)\}$$...
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1answer
44 views

Taking a derivative of a function with respect to another function

I read a set of notes recently (unfortunately I can't find the link) in which the author made a statement of the form "differentiation of a function with respect to a function doesn't make sense". By ...
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1answer
28 views

Derivative of unknown compound function

The problem says: What is $f'(0)$, given that $f\left(\sin x −\frac{\sqrt 3}{2}\right) = f(3x − \pi) + 3x − \pi$, $x \in [−\pi/2, \pi/2]$. So I called $g(x) = \sin x −\dfrac{\sqrt 3}{2}$...
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0answers
28 views

Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where     &...
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1answer
25 views

Image and Kernel of composition of two homomorphisms

I have just showed that the composition of $a * b$ of two homomorphisms $a,b$ is a homomorphism. However, what can I say about the image and kernel of $a*b$, in terms of $a$ and $b$? Is there ...
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1answer
16 views

Function Composition, Derivatives, Gradient, Hessian

Here's the problem: Let $f : R^n \to R$ be a twice continuously differentiable function. Let $\phi(t) = f(u + td)$ be a composition function from $R$ to $R$, with given vectors $u, d \in R^n$. ...
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0answers
24 views

M is a k-manifold if and only if $\phi(M)$ is a k-manifold

Let $\phi: \mathbb{R}^n\rightarrow \mathbb{R}^n$ be a diffeomorphism and $M\subset \mathbb{R}^n$ M is a k-manifold if and only if $\phi(M)$ is a k-manifold. Prove it. So what I did was try to ...
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27 views

convexity of composite function

There is a composite function F(x)=f(g(x)): R -> R where f(v): R^2 -> R and g(x): R -> R^2. It is given that f(v) is convex on v and v = g(x) gives a map R -> R^2. If x is in a convex set, is F(x) ...
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1answer
32 views

Proving that a function is an isomorphism of groups

Let A and B be non-empty sets and f : A → B be a bijection. Consider the map $\phi$ : $S_A$ → $S_B$ that sends $\sigma$ to ${f} \circ {\sigma} \circ {f^{-1}}$. Show that $\phi$ is an isomorphism of ...
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1answer
26 views

Composition of three quadratic functions

Is it possible to find three quadratic functions $f(x),g(x)$ and $h(x)$ such that $f(g(h(x)))$ has $-6,-5,-4,-2,1,3,4,5$ as its roots? I understand that the composition of three quadratic functions ...
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0answers
57 views

How to define composition of distribution with a function correctly?

Recently I've been reading some notes on distribution theory and the author makes the following definition: Let $\zeta\in \mathcal{D}'(\mathbb{R})$ be a distribution and $f$ a $C^\infty$ function, ...
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1answer
41 views

Compostion on $H^2(U)$

Below is a question that I'm attempting to do but so far have made no progress. Any suggestions would be helpful. Show that whenever $0 < \alpha < \frac{1}{2}$, then $\left( \frac{1+z}{1-z} \...
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2answers
118 views

are there any polynomial-exponential, bell-shaped functions? [closed]

I am looking for a polynomial-exponential, bell-shaped function under the restrictions below. Definition: By polynomial-exponential function I mean something of the sort $g(x)^{h(x)}$ where $g(\cdot)...
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1answer
58 views

Singularities of Composition of Functions

We are learning about singularities in my Complex Analysis course right now. I understand what it means to be each type of singularity however, a question I had was about the singularities of ...
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1answer
22 views

Domain of composite functions

Given $f(x)= 1- x^2$ and $g(x) = \sqrt x$ What is the domain of $f$ and $g$? My answer is that the domain of $f$ is all real $x$, and the domain of $g$ is all $x \ge 0$. However, I am not sure if ...
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0answers
16 views

Antitone Galois connections, composition

Is composition of two antitone Galois connections defined? What are all "possible" ways to define composition of two antitone Galois connections?
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2answers
4k views

Given $g(x)$ and $f(g(x))$, solve for $f(x)$.

I've hit a wall on the above question and was unable to find any online examples that also contain trig in $f(g(x))$. I'm sure I am missing something blatantly obvious but I can't quite get it. $$ g(...
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1answer
18 views

Using function compositions to determine variable values

Let $f,g:$ $R \rightarrow R$ where $g(x)$ $=$ $1$ $-$ $x$ $+$ $x^2$ and $f(x)$ $=$ $ax$ $+$ $b$ If $g(f(x))$ $=$ $9x^2$ $-$ $9x$ $+$ $3$, determine $a$ and $b$. So far, I have "fit" $f(x)$ into $g(x)...
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2answers
29 views

How to do compositions of functions

I've been given a problem that says to find $[f \circ g](x)$ and $[g\circ f](x)$. First off, what is the difference between the two? Second, how might I do this?
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1answer
26 views

An Injective Composition of Linear Transformation

Suppose that A is a linear transformation from vector spaces U to V and that B is a linear transformation from vector spaces V to W. Suppose further that B composed of A is an injective composition of ...
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1answer
62 views

Proof involving functions

Prove that $q(n) = n$ is the only onto function that satisfies $q \circ p = p \circ q$
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1answer
69 views

Proving that if two linear transformations are one-to-one and onto, then their composition is also.

I am attempting to solve a problem with the following given conditions: Let V, W. and Z be vector spaces, and let $T:V \longrightarrow W$ and $U: W\longrightarrow Z$ be linear.Prove that if U and T ...
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1answer
40 views

Finding a transformation that yields a prescribed PDF

I am attempted to procure a function from a composition when given the PDF (I typed the full problem at the bottom in its entirety in case I left out details in my inquiry). I understand how to get ...
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1answer
44 views

Find the values of $k$ for which the equation $(f\circ g)(x) = x$ has two equal roots

I'm busy doing a problem which asks the above considering the following: $$f(x) = 4x - 2k\text{ and }g(x) = 9/(2-x)$$ As far as I know roots usually refer to quadratics and even when doing the ...
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0answers
50 views

Solving a delay differential equation

Is it possible to solve differential equations with composite functions, e.g. $f^{(n)}\circ (x-1)$ and $f^{(n)}\circ(x)$. I'm particularly interested in $$2f'(x-1)+(x-1)f''(x-1)-f''(x)=g(x).$$ Do ...
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0answers
17 views

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 ...