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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15 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 ...
4
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4answers
79 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
18 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 \\ ...
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2answers
92 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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0answers
48 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 ...
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0answers
35 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
23 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
18 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 ...
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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 = ...
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1answer
38 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
26 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 ...
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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
23 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
14 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 ...
0
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0answers
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
25 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 ...
2
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0answers
54 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
115 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 ...
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1answer
44 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
15 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. $$ ...
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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 ...
0
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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
25 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
68 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
39 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 ...
0
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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
46 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 ...
0
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0answers
15 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 ...
2
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3answers
46 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 ...
4
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1answer
29 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 ...
0
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1answer
22 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
17 views

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 ...
2
votes
1answer
76 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.
2
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1answer
40 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 ...
0
votes
1answer
27 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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votes
2answers
289 views

How does the continuity of a composite function of floor change when it is integrated while being held fixed? [closed]

Now we know that the integral and derivative work with functions a certain way and that they treat them in some unique. Now, for the rest of this question let's just act a little weird here and ...
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0answers
16 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 ...
2
votes
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 ...
0
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0answers
16 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
21 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 $g: Y \to Z$.
0
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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 ...
3
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0answers
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 $\mathbb{R}$. ...
0
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1answer
12 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 ...
0
votes
4answers
74 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 $f$?