Questions tagged [function-and-relation-composition]

Question 1

Let $A \subset \mathbb{R}$ be a finite set with $|A| = n$ and let $f : A \to A$ satisfy the strict contraction condition $|f(x) - f(y)| < |x - y|$ for all $x \neq y$ in $A$. Prove that $f$ is not ...

Question 2

I'm currently studying tensor algebra and analysis in $\mathbb{R}^3$ because I need it for continuum mechanics purposes and I'm currently focusing on 2-nd order tensors. The vector space is Euclidean ...

Question 3

I am having difficulties to formally prove that, in the derivative of composition of two functions $g$ and $f$, the requirement that $f(D_f) \subseteq D_g$ (where $D_f$ and $D_g$ are intervals and ...

Question 4

We use the definition of elementary functions given in Spivak's calculus (with some changes so this is not exactly the same). An elementary function is one which can be obtained by a finite number of ...

Question 5

Let $f(x) = 2x^2 + x - 1$ and $$ f^n(x) = (\underbrace{f \circ \dotsb \circ f}_{n \text{ copies of } f})(x). $$ Given $n \ge 1$, what is the number of real roots to $f^n(x) = 0$? Note: This is an ...

Question 6

The problem is: Find the number of real solutions of $f^n(x)=0$ where $f(x)=2x^2+x-1$. ($f^1(x)=f(x)$, $f^n(x)=f(f^{n-1}(x))$.) I tried drawing graph and counting the zeros, and I got the conjecture ...

Question 7

I have three topological space, $\Omega$, $R$, and $S$. Then consider function spaces $\bigotimes_{\omega \in \Omega} R$ (here $\bigotimes$ is a generalized Cartesian product; not my prefered $\LaTeX$ ...

Question 8

Found on AoPS $$ \text{Let } f(x) = x^3 - \frac{3}{2}x^2 + x + \frac{1}{4}. \text{ For every } n \in \mathbb{N} \text{ let } f^n \text{ denote } f \text{ composed } n\text{-times, i.e.,} $$ $$ f^{n}(...

Question 9

For a suitably chosen real constant $a$, let function $f:\mathbb{R}-\{-a\}\to \mathbb{R}$ be defined by $$f(x)=\dfrac{a-x}{a+x}$$ Further suppose that for any real number $x\neq -a$ and $f(x)\neq -a$, ...

Question 10

Matrix Calculus utilizes function composition of, at least phenomenologically, several different types. While I've encountered each before in different subjects, synthesizing them all into the same ...

Question 11

(1)Let $f,g :[0,1] \rightarrow \mathbb{R}$ be continuous functions such that for $a,b\in [0,1]$ , we have $f(a)=f(b) \implies g(a)=g(b)$ Show that there exist a continuous map $h:f([0,1]) \rightarrow \...

Question 12

I was wondering if say for increasing functions $f(x),g(x)$ with $f(x)$ asymptotically growing faster, $f(g(x))$ grows faster than $g(f(x))$. As is, I know you cannot say as you can find examples ...

Question 13

Edit: $f, g_n, g$ are diffeomorphisms. There are several instances of this question for the uniform topology, but I'm concerned at least with the case in which $f$ and $g$ are bi-Lipschitz ...

Question 14

I was reviewing Khan Academy's composite limit theorem and tried to cross reference it to other sources to understand it better. I understand it less now. I found out that there is at least two ...

Question 15

I'm having troubles in understanding a rapidly solved exercise, which is actually a sort of chain of consequences. It says: be $A; B$ two matrices $n\times n$ over a field $K$ and be $F_A, F_B$ their ...

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Questions tagged [function-and-relation-composition]

Prove that the iterate $f^{n}$ is a constant function.

Powers and roots of 2-nd order tensors in Euclidean space

Why, if we drop $f(D_f) \subseteq D_g$ for $f(a) \in D_g$, then chain rule can't hold?

Is the condition about composition needed to stay with the elementary functions?

The number of real roots of $f^n(x) = 0$ where $f(x) = 2x^2 + x - 1$

The number of real solutions of $f^n(x)=0$ where $f(x)=2x^2+x-1$. [closed]

Is the composition of two continuous functions a continuous operation?

Let $f(x) = x^3 - \frac{3}{2}x^2 + x + \frac{1}{4}$. Evaluate $\int_0^1 f^{2025}(x)\, dx. $

How to identify incorrect value of $a$

Harmonizing composition in Matrix Calculus

Existence of a continuous function $h$ such that $g=h\circ f$ given $f$ and $g$

Is there any way to compare growth of function compositions?

If $f, g_n, g = \lim_n g_n \in C^1(M, M)$, where $M$ is a compact manifold, does $\lim_n f \circ g_n = f \circ g$ in the $C^1$ topology?

Which Composite Limit Theorem to Trust?

Linear transformations and their composition: why is this an isomorphism? [SOLVED]

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