# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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