Questions tagged [function-and-relation-composition]
For questions about the composition of functions and relations.
1,234 questions
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Powers and roots of 2-nd order tensors in Euclidean space
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 ...
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Why, if we drop $f(D_f) \subseteq D_g$ for $f(a) \in D_g$, then chain rule can't hold?
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 ...
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Is the condition about composition needed to stay with the elementary functions?
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 ...
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The number of real roots of $f^n(x) = 0$ where $f(x) = 2x^2 + x - 1$
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 ...
- 3,145
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The number of real solutions of $f^n(x)=0$ where $f(x)=2x^2+x-1$. [closed]
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 ...
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Is the composition of two continuous functions a continuous operation?
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 $\LaTeX$ ...
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Let $f(x) = x^3 - \frac{3}{2}x^2 + x + \frac{1}{4}$. Evaluate $\int_0^1 f^{2025}(x)\, dx. $
Found on AoPS
$$
\text{Let } f(x) = x^3 - \frac{3}{2}x^2 + x + \frac{1}{4}. \text{ For every } n \in \mathbb{N} \text{ let } f^n \text{ denote } f \text{ composed } n\text{-times, i.e.,}
$$
$$
f^{n}(...
- 3,478
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How to identify incorrect value of $a$
For a suitably chosen real constant $a$, let function $f:\mathbb{R}-\{-a\}\to \mathbb{R}$ be defined by $$f(x)=\dfrac{a-x}{a+x}$$
Further suppose that for any real number $x\neq -a$ and $f(x)\neq -a$, ...
- 1,074
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Harmonizing composition in Matrix Calculus
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 ...
- 2,184
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Existence of a continuous function $h$ such that $g=h\circ f$ given $f$ and $g$
(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 \...
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4
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Is there any way to compare growth of function compositions?
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 ...
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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?
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 ...
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Which Composite Limit Theorem to Trust?
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 ...
1
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1
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Linear transformations and their composition: why is this an isomorphism? [SOLVED]
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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If $f(x) = x^3$ and $g(x) = x^4$, prove that $f \circ g = g \circ f$.
If $f(x) = x^3$ and $g(x) = x^4$, prove that $f \circ g = g \circ f$. ($x \in \mathbb{R}$ and $f: \mathbb{R} \rightarrow \mathbb{R}$, $g: \mathbb{R} \rightarrow \mathbb{R}_+$)
The solution is simple:
...
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