Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

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Find $f''(2x)$ if $f'(x) = g(x + 1)$ and $g'(x) = h(x - 1)$

Find $f''(2x)$ if $f'(x) = g(x + 1)$ and $g'(x) = h(x - 1)$ Hello, I am stuck on the above problem. Here's my work: Differentiate once: $$\frac{d}{dx} (f(2x)) = 2 \cdot f'(2x) \text{ (Chain Rule)}$$ $...
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Difference between superset and universal set

1- Let A={a,b} and B={a,b,c,d,e}. I know that B is superset of A. But it seems a universal set as well. 2- If A={a,b}, B= {a,b,c,d,e} and C= {c,d}. B is still a superset of A and C. Is it also a ...
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Inverse of $f(x)= x^2$, under $\mathbb R\rightarrow \mathbb R$.

Find Inverse of $f(x)= x^2$ under the operation of function composition $\circ$, with domain and codomain being $\mathbb R$, i.e. $\mathbb R\rightarrow \mathbb R$. The given function fails to have ...
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Is there a non-analytic function $g$ such that $\exp(s g(z))$ is always analytic?

Does there exist a complex function $g(z)$ that is not analytic on the unit disk, such that $\exp(s g(z))$ is analytic on the unit disk for all complex numbers $s$? What about for some continuous ...
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If we know the graph of $f(x)$ and of $g(x)$, is there a way to graph their composition $f(g(x))$?

My question is that if we know the graph of $f(x)$ and of $g(x)$, s there a way to graph $f(g(x))$ Example: $\sin (\ln (x))$ How do we reach this graph? How does this graph relate to its parent ...
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Does the simplicity lemma of Koenig's theorem applied to $t-\frac{f(t)}{\frac{d}{dt}f(t)}$ indiciate the simplicity of the roots of f?

Koenigs Linearization Theorem: If the magnitude (absolute value) of the multiplier $\lambda = \dot{f} (0)$ of a holomorphic map $f$ is not strictly equal to 0 or 1, that is $λ \ne \{0, 1 \}$, then a ...
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Is the composition of two optimal transport maps still optimal (under some assumptions)?

Consider three absolutely continuous probability measures $\mu$, $u$, and $\nu$ on $\mathbb R^d$ ($d \geq 1$), all of which have finite second moments. A transport map from $\mu$ to $\nu$ is called ...
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computational complexity of a function involving matrix operations

I am trying to understand computational complexity in evaluation of certain class of functions on MATLAB. I wanted to just verify if I am doing this correctly. I have the following function, $f: \...
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Why $f^2(x) \ne f(x)^2$?

I am working on an exploration which starts with the following affirmation: In this section you studied the Binomial theorem. Recall function composition from earlier in the course. In this context (...
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Function class consisting of gradients of real-valued convex functions

Denote $\mathcal F$ as the function class consisting of gradients of all real-valued convex functions in $\mathbb R^d$, that is, $\mathcal F = \{ \nabla \phi ~|~ \phi: \mathbb R^d \to \mathbb R \text{ ...
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If $𝑓∘𝑔∘ℎ=𝑓 ∧ 𝑔∘ℎ∘𝑓=𝑔$ then must $ℎ∘𝑓∘𝑔=ℎ$?

If not, then What can be said of each $𝑓,𝑔,ℎ$ and are there any simpy-definable minimal conditions imposable upon one or more of the indexable functions that would ensure this symmetric closure? ...
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Why does Jim Hefferon use the codomain, instead of the range, for defining inverse functions?

In the book "Linear algebra" specifically on Appendix page A-8. "In line with that analogy [how 0 is the addition identity and 1 is the multiplication identity], we define a left ...
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What kind of operation is cube root extraction?

I came across this question in a random test and the correct answer was marked as "Binary Operation". I am pretty sure that to find the cube root of a number you only need that number alone ...
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Find an unknown function, with a given function result as a composition of its inverse with another function

Given the injective functions $f$ and $g$, defined by: \begin{equation} f(x)= \begin{cases} 2-x^2 & \text{if } x \in [\sqrt{3},2]\\ 1-\sqrt{x^2-4} & \text{if } x \leq-4 ...
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Composition of a relation with its inverse

I'm self-learning my way through Set theory and came across this question. Now I have a few difficulties to gain an entry to this question. The textbook and the lecture videos which I'm using gives ...
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Composite Function Domain will include infinity or not in the below case

Consider the function h(x) = $f(g(x)) = \frac{3}{\frac{6}{3-x}}$ from $g(x) = \frac{6}{3-x}$ and $f(x) = \frac{3}{x}$ , we will say that h(x) to be not defined at infinity and -infinity as both those ...
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Condition on composition functions for the given fact to be true in case of codomain off is not same the domain of g

We know that for $f:Y \to Z$ and $g:X \to Y$ , if $f \circ g$ is either bijective or just surjective or injective we have $g$ is injective and $f$ is surjective for first case, similarily $f$ is ...
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Decompose a function

I have $f(x^2-x)=x$ and I would like to find $f(x)$. Is there a systematic way to do it, which also works for similar composite functions?
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Is composition of reversible functions reversible too? [duplicate]

This sounds obvious. Too obvious for me to be able to prove it formally. Can you please help with a formal proof of this?
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Using the following definitions for functions and their compositions, how do I prove that the composition of two functions is also a function?

Definition of a function (i.e a set $f$ of ordered pairs is a function, if and only if): $f:A\rightarrow B \iff \forall x(x \in f \implies x \in A \times B) \land \forall x(x \in A \implies \exists! y:...
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Decomposing higher order derivatives of composite functions

I'm new to calculus, and just a little hazy on the skeleton of higher order derivatives when the chain rule is involved. I'm given a table of function and first derivative values, and I need to solve ...
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How can one find solutions to $f''(x)=f(f(x))$?

This was a differential equation that I came up with. I have never seen any ODEs which involve composition so I have no idea on how to approach this. One solution appears to be $f(x)=0$ but I can't ...
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Example of $f,g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is bijective but $g\circ f$ is not bijective

I am seeking functions $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ such that $f\circ g$ is a bijection and $g\circ f$ is not a bijection. Here is what I have done so far in terms of ...
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Are the theorems g•f injective/surjective then f/g is injective/surjective invertible?

i have questions. I know that: if $g \circ f$ is surjective than g is surjective; If $g \circ f$ is injective than f is injective; If g and f are both injective than $g \circ f$ is injective; If g and ...
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Am I using the right formal logical statement to represent the relationship between two sets?

Given two sets named bigger_set andsmaller_set with an element of smaller_set "pointing&...
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Relations and Cartesian Product: why does $X^1$ differ from $X^0$ and $X^2$?

Question Why are functions of a single variable specified with a tuple argument $(x)$ as in $f(x)$, when this pattern suggests they should be specified without parenthesis as $fx$? This pattern (1): $$...
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$L: \text{Aff}\left ( \mathbb{R}^n ,\mathbb{R}^n \right ) \to M(n,\mathbb{R})$ is a map of rings?

First of all, $\text{Aff}\left ( \mathbb{R}^n ,\mathbb{R}^n \right )$ is a set of affine maps $\mathbb{R}^n \to \mathbb{R}^n$. And, $M(n,\mathbb{R})$ is a set of all $n \times n$ matrices over $\...
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Any Glide is a product of a Half turn and a Reflection. I have a solution, but I can't understand it.

The plane we are dealing with is $\mathbb{R}^2$. Notations: $H_a = R(a,\pi)$ which means a rotation about the point $a$ for $\pi$. $G(l,a)=R_lT_a$ which means a glide that is a composition of a ...
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Composition of two rotations with angle $\alpha$ and $-\alpha$, but with different centers is a translation.

I would like to get some explanations for this fact. (The plane is $\mathbb{R}^2$) So, I've been drawing on a grid to see what a composition of $R(a,\alpha) R(b,-\alpha)$ (but, $a \neq b$) gives us. ...
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Any translations can be written in a composition of two half-turns. But, how do we show this?

On the plane $\mathbb{R}^2$, I want to show that any translation can be written as a composition of two half-turns $H_a, H_b$ with either $a$ or $b$ chosen arbitrarily. I know that any composition of ...
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Continuous function composed with itself is equal to propagation of a differential equation.

This question has been bugging me for a while. It was given as the last question of a first year undergrad analysis exam and so should be solvable with little machinery, yet it seems to point straight ...
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Moving a center of a rotation. (Decomposition of a rotation)

We denote: $R(a,\alpha)$ :"rotation about point $a$ by angle $\alpha$. $T_a$ : "translation" (e.g: $T_a(x)=x+a$). I'm trying to simplify the composition of two rotations $R(a,\alpha)$ ...
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Find the composition of two relations Rp and Rq

Let $S_{n}$ be the set of all n-element sets in some universe. Given that $R_{k}$ is the relation on $S_{n}$ which is defined as x$R_k$y if and only if the intersection of x and y has at least k ...
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Suppose that $h : X → Y$ and $f : Z → Y$. Then $\exists g : X → Z$ s.t. $h = f ◦ g$ if $f$ is a bijection.

Suppose that $h : X → Y$ and $f : Z → Y$. There exists a function $g : X → Z$ such that $h = f ◦ g$ if $f$ is a bijection. I am not sure how to go about proving this. Please tell guide me. proof: I ...
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Why are mappings notated LTR despite compositition being RTL?

Say you have the general mappings $f: S\rightarrow T$ and $g: T\rightarrow U$. Wouldn't the notation $f: T\leftarrow S$ and $g: U\leftarrow T$ be more consistent with the right-to-left nature of ...
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On the representation of composition of functions: how to pronounce $g\circ f$ when speaking? [duplicate]

Let $f:X \rightarrow Y$ and $g:Y\rightarrow Z$ Then one defines composition of functions as follows, $$g(f(s)):X\rightarrow Z $$ For all $s\in X$. I also saw the following notation being used, $$g \...
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Composition of piecewise functions - Strange result

I'm trying to get $f(g(x))$, where: $$ f(x)= \begin{cases} \sqrt{1-x} &\text{if } x \leq 1 \\ x &\text{if } x > 1 \end{cases} $$ $$ g(x)= \begin{cases} x + 1 &\text{if } x \geq 0 \\ x^2 ...
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Composition of algebraic functions is algebraic

Let $f\colon D \subset \mathbb{R} \to \mathbb{R}$ and $g\colon E \subset \mathbb{R} \to \mathbb{R}$ be algebraic functions such that $f(D) \subset E$. I want to prove that $g \circ f\colon D \to R$ is ...
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Mapping Identity Proof [closed]

I'm struggling with proving the following: Let $A$ be a set such that $|A| \ge 2$ and let $f: A \to A$ be a mapping on $A$ such that $f(g) = g(f)$ for all mappings $g: A \to A$. Prove that $f$ is the ...
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3 answers
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Why does taking the square root to the independent variable (x) cause the squaring of x in each point?

Why does the graph of $\text{sin}(\sqrt{x})$ look like this why is the maximum at ($(\frac{\pi}{2})^2,1)$ and not ($\frac{\sqrt\pi}{\sqrt2},1)$ why did we square and not take the square root and does ...
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Limit Laws for Composite Functions

There are two rules I've seen for composite Limits. The first one is: If $f(x)$ is continuous at $x=b$ and $\newcommand{\limto}[2]{\lim\limits_{{#1}\to{#2}}}\limto xa g(x)=b$ then, $$\limto xa f(g(x)...
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What technique, method or procedure exists for someone to determine a recurring or explicit function that fits a set of data points?

I have an infinite set of data points (the first seven values are posted bellow) and I'm trying to figure out how someone would go about determining the recurring or explicit function from them? ...
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Composition of relation R = E9 ◦ E12

Given the relations E9 and E12 on the set of integers defined as follows • aE9b precisely when a ≡ b (mod 9). • aE12b precisely when a ≡ b (mod 12). Let R = E9 ◦ E12 (a) Suppose that 2Rx. Prove that x ...
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Composing with a function + constant

Consider the discounting condition in the Blackwell's sufficient conditions: (Reference: A related question). There exists some $\beta \in (0, 1)$ such that $[T(f + a)](x) ≤ (T f)(x) + βa$, for all $f ...
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Questions Regarding this Chain Rule Proof

I saw this proof of the Chain Rule on Hardy's A Course of Pure Mathematics (the notation I use will be a little different). • Chain Rule: Let $f\,\colon Y \subset \mathbb{R} \to \mathbb{R}$, $g\,\...
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Are there two kinds of n-fold composition? Left and right?

Let $f^{\circ n}$ denote the $n$-fold composition of function $f$. As an example, $f^{\circ 3}x$ is short-hand for $f(f(fx))$. Is there another form of composition to denote $((ff)f)x$? Would one be “...
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How to find specific functional square roots (like half iteration of exp, ln, ...)?

Curious about intermediate mean between arithmetic and geometric, I noticed that all follow pattern $M(\space f \space,\space x[1,2,...,n] \space,\space w[1,2,...,n] \space)=f^{-1}(\frac{w[1]\space*\...
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Is it necessary for the codomain of the inner function to be equal to the domain of the outer function in function composition?

Recently I came across a statement which meant roughly the following. If we want to apply composition of functions, then the co-domain of the inner function should be equal to the domain of the outer ...
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Inverse of a composite function

A standard property of composite functions is that: $$(f\circ g)^{-1}(z)=(g^{-1}\circ f^{-1})(z)$$ Is it acceptable to prove this property at the level of a high school proof (not college level), by ...
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1 vote
1 answer
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Proving bounded gradient for composite function

I have a non-convex bounded function $h$ that is L-smooth and it has bounded gradient, i.e. $\|\nabla h(x)\| \leq \sigma^2, \forall x \in \mathbb{R}^d$. Define the function $f(y) = \exp(\alpha \times ...
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