Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

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26 views

Translation of the function in Complex

Consider the $T : z \to z+a$ for $a \in \mathbb{C}$ (Here the $a = \alpha + i \beta$) Take any complex function, $f(z)$ Then, $T \circ f(z) = f(z) + a$ is translation by $a$ (I.e. $f(z)$ is translated ...
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Let 𝑎, 𝑏 ∈ ℝ and let 𝑓$_{𝑎,𝑏}$: ℝ → ℝ be the function defined as 𝑓$_{𝑎,𝑏}$ (𝑥) = 𝑎𝑥 + 𝑏 for all 𝑥 ∈ ℝ. Use this for the problems below.

a) Prove that 𝐺 = {𝑓$_{𝑎,𝑏}$ | 𝑎, 𝑏 ∈ ℝ, 𝑎𝑛𝑑 𝑎 ≠ 0} is a group, where the operation is composition. Let 𝐺1 = {$\big(\begin{smallmatrix} a & 0\\ b & 1 \end{smallmatrix}\big)$ ∶ �...
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Confusion regarding symmetric and anti-symmetric relation

let R be a relation on a collection of sets defined as follows, R { ( A , B ) | A ⊆ B } Here, since it is an improper set, can't we take it as A=B and say that it is both symmetric and anti-symmetric ...
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3answers
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How can I prove that a function is invertible, where do I get the function and the necessary data?

"If g is another invertible function, then the compound function $f ∘ g$ is also invertible, and it is fulfilled.." $$( f ∘g) ^{-1} = g^{-1} ∘ f^{-1}$$ I don't even understand where to start....
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1answer
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How do I prove or disprove these relation compositions? [closed]

So I am not told if these statements are true or not (although they appear true). I am asked to either prove them in general terms, or disprove them with specific examples. I am confused about ...
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1answer
16 views

Composition of a Realtion with identity is the relation itself

I am completely lost as to how to prove the following, although it appears straightforward. I know how this holds for functions, but how do I prove it specifically for relations? I need to provide a ...
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Continuity possible?

Can functions $f$ and $g$ not be continuous at $8$ while function $f$ of $g$ is continuous at $8$? So far I've tried using piecewise functions to accomplish this but haven't been successful.
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1answer
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How to calculate the maximal domain and hence range in this question?

Question: The function $f : R → R, f (x)$ is a polynomial function of degree 4. Part of the graph of $f$ is shown below. The graph of $f$ touches the x-axis at the origin. part a) wants me to find ...
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3answers
79 views

Let $f(x) = |x+1|-|x-1|$, find $f \circ f\circ f\circ f … \circ f(x)$ (n times).

Let $f(x) = |x+1|-|x-1|$, find $f \circ f\circ f\circ f ... \circ f(x)$ (n times). I don't know where to start... Should I use mathematical induction? But what should be my hypothesis? Should I ...
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1answer
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Calculating Composition of 2 Functions

Let $f: \mathbb{R}^2 \to \mathbb{R} $ and $g: \mathbb{R} \to \mathbb{R}^2$. Suppose $g(0) = (1,2)$, $g'(0) = (-1,4)$ and $(f \circ g)'(0) = -3 $. Furthermore $\displaystyle \frac{df}{dx}(1,2) = \frac{...
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Functional equation involving composition and exponential

My main question is how to solve this functional equation: $f(f(x)) = e^x$. The context within which this came up was when I was attempting to extend the notation of the inverse function ($f^{-1}$) to ...
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1answer
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Finding the domain of a composition of functions where one function is given in terms of y and the other in terms of x.

For a math assignment I was assigned by my professor, I've been asked to find the domain of a composition of the following functions. $f(y) =\frac{4}{y - 2}$ $g(x) =\frac{5}{3x - 1}$ I know that the ...
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1answer
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Composite functions with more complicated functions

I have this question, as below, and I cannot understand what I am being asked of. Typically I would consider the composition of functions to be rather simple, such as $f(x)=2x-5$ $g(x)=x-2$, therefore ...
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1answer
26 views

How to See if a Composite Function is One-To-One and Onto

So we have $f$ that is onto, and $g$ which is onto and one-to-one. Is $f \circ g$ onto and one-to-one? My attempt at to solve this problem was to state the following scenarios and come to a conclusion....
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1answer
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Prove that any map $f : X \to Y$ can be represented as composition of 1) inj and surj. 2) surj. and inj.

I was given the following task Prove that any map $f : X \to Y$ can be represented as composition $f = g \circ h$, where $g$ is a surjection and $h$ is injection composition $f = z \circ c$, where $...
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3answers
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Why $f(g(x)) = x$ and $f(x)=g(x)$ imply $f(x)=x$?

I was watching this video from blackpenredpen where he solves the equation $\sqrt{5-x}=5-x^2$ by writing it in terms of "5". However, there's a comment with an alternate solution using ...
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1answer
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How to determine the nature of two functions from their composite function?

The following question was asked in an examination: Let $f$ and $g$ be two functions with domain and codomain equal to the set of real numbers. If, $$g\circ f(x) = \begin{cases} x^2, & \text{if $...
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1answer
23 views

Correct Notation for Limits of Function Composition

When taking the limit of function composition, what is the best way to write the intermediate step? Suppose we have two functions $f(x)$ and $g(x)$ that are continuous where $\lim_{x\to a} g(x)=b$ and ...
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1answer
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Proving this continuous function is entire if a related function is given entire

This question was asked in a masters exam for which I am preparing and I was unable to think about it. So, I am asking it here. Question: Let f be a continuous function from $\mathbb{C} \to \mathbb{C}...
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1answer
38 views

How is it possible for the composite function to be continuous in this case?

When $f(x) = x^2+3x+1 (x\geq1), x+2(x<1)$, $g(t) = ㅣ3t-12ㅣ$, $t=f(x)$ In this case, how can $g(f(x))$ be define between when $f(x)= [3,5]$? $f(x)$ do not have $3<f(x)\leq5$ as a result. so $g(t)$...
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If $f(x)=\frac{1-x}{1+x}$, then how do I explain graphically why $f(f(x))=x$?

Let $f(x)=\dfrac{1-x}{1+x}$. Then $f(f(x))=x$. The domain of $f(f(x))$ is $\mathbb{R}$\ ${-1}$ How do I explain graphically why the formula for $f(f(x))$ is the way it is?
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2answers
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Odd and even properties of functions

Could I get a hint or a clue on how to solve this problem? Problem: ($f$ is even if $f(-x)=f(x)$ and $f$ is odd if $f(-x)=-f(x)$.) Suppose $f$ is a function defined on all of $\mathbb{R}$ (a) Check ...
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2answers
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Is $x-y \ne 0$ a transitive relation?

I know that for a relation R is transitive if, for all elements aRb and bRc implies aRc. I came across this question, " Is x-y ≠ 0 transitive?" I think it is transitive since x - y ≠ 0 y -...
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0answers
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Composition of jets

I was reading that for a function $f:\mathbb{R}^m\rightarrow \mathbb{R}^n$ such that $f(0)=0$ and a function $g:\mathbb{R}^n\rightarrow \mathbb{R}^d$, $J(g\circ f)=J(g)\circ J(f)$ where $J(f)$ denotes ...
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Chain rule of higher derivative with multivariable functions

I wanted to know if there is a general formula in order to calculate $\frac{\partial^{k}f\circ g}{\partial x_{i_k}\dots \partial x_{i_k}}$ using $\frac{\partial^{d}f}{\partial x_{j_d}\dots \partial x_{...
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1answer
19 views

Composition of an inverse function with another function

Suppose I have two functions $f(x)$ and $g(y)$. Then what is $f^{-1}$ composed with $g$, i.e.: $f^{-1}\circ g$ ? To me it looks like the value of $x$ when $f(x) = g(y)$, but I am not entirely sure. ...
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1answer
31 views

Function composition on a commutative diagram: basic question

I'm trying to work through Categories for the Working Mathematician and immediately ran into a point of confusion. A composition of functions is commutative iff $f\circ g = g\circ f$. But if in this ...
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1answer
30 views

An extra condition for composition of relations which are not necessarily functions

Given two binary relations $\mathcal R$ and $\mathcal S$ over $A \times B$ and $B \times C$,then their composition relation is denoted by $\mathcal S \circ \mathcal R$ and is defined as: $$\mathcal S \...
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1answer
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Finding the Integral of A Composition of Logarithm, exp, and Trigometric Functions

I have the function $$f(x)=\ln \left(e^{5+6\left(\sum_{k=0}^{100} \frac{\sin \left(k\pi x\right)}k \right)}+e \right)$$ One, is it possible to integrate this without "special functions" (...
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2answers
60 views

Finding a variable $n$ that satisfies the functional composition in which $f(f(f(f(n))))=3.$

Let $f(x) = x^2-2x$. How many distinct real numbers $c$ satisfy $f(f(f(f(n)))) = 3$? I have not found any good way to do this problem. I have just resorted to start with brute force: We start with $$...
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2answers
56 views

A binary relation $\mathcal R$ over a set $A$ is transitive if and only if $\mathcal R$ is equal to its transitive closure $\mathcal R^{+}$.

Given a binary relation $\mathcal R$ over a set $A$,then the $\mathsf {Transitive \;Closure}$ of $\mathcal R$ over $A$ is the smallest transitive relation on $A$ containing $\mathcal R$, it's indeed ...
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2answers
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Conjecture about the transitive closure of a relation $\mathcal R $ over a finite set $A$

Given a binary relation $\mathcal R$ over a set $A$,then the transitive closure of $\mathcal R$ over $A$ is the smallest transitive relation on $A$ containing $\mathcal R$, it's indeed the ...
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0answers
49 views

Partial derivatives of a multivariable composite function (chain rule)

Say I have this function: $$z=f(ax, ay)$$ I know how to take this partial derivative by defining $u=ax$ and $v=ay$: $$\frac{\partial z}{\partial x}=\frac{\partial f(ax, ay)}{\partial x}=\frac{\partial ...
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1answer
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Decomposition of a function and chain rule.

This question is about the basic chain rule (and I think of it when I read about calculation of variation in defining distance in manifold using usual Riemannian metrics) and is related to the another ...
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1answer
58 views

Can we really compose random variables and probability density functions?

A renowned professor of statistics (whose name I will not reveal here) told me that the notation $p(x)$ makes perfect sense when $p$ is a pdf and $x$ is a RANDOM variable (i.e. a function). I was a ...
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1answer
26 views

Composition Functions (Advanced Functions)

Question: a) Given the functions $f(x) = x + 2$ and $g(x) = 3^x$, determine an equation for (f ∘ g)(x) and (g ∘ f)(x). b) Determine all values for $x$ for which $f(g(x)) = g(f(x))$. *For part a), I ...
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Composing trigonometric functions

Let $f(x)=\sin(x)$. If $g$ and $h$ are functions on $\mathbb{R}$ such that $g(f(x))= h(f(x))$, can we conclude $g=h$ ? Can we actually compare $g$ and $h$? I am confused. Please, help me.
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1answer
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Questions about relation composition and equivalence relation

Let $R \subseteq S\times S$ be any binary relation on a set S. Consider the sequence $R^{0}, R^{1}, R^{2}...$ defined as follows: $$R^{0} := I = \left \{ (x,x):x\in S \right \}\\ R^{n+1} :=R^{n}\cup \...
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1answer
29 views

Continuity with composition functions

Is there a continuty in $g(f(x))$ at $x = 4$? $$f(x)=\begin{cases} -1 &, x=4 \\ 1 &, x \ne 4\end{cases}$$ $$g(x)=\begin{cases} -6 &, x=4 \\ 4x-10 & x \ne 4\end{cases}$$ My answer for ...
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1answer
85 views

Is $f^2 \circ f=f \circ f^2$ true?

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a function from the set of real numbers to the same set with $f(x)=x+1$. We write $f^{2}$ to represent $f \circ f and f^{n+1}=f^n \circ f$. Is it true ...
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1answer
20 views

Need the result of composing an infinite number of smooth functions be smooth?

$f$ is a smooth function from a manifold to itself. So is $f\circ f$, and $f\circ f\circ f$ and so on... If this sequence is extended forever, and supposing that it converges to some function, need ...
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1answer
79 views

Computing $f \circ g$ and $g \circ f$ for functions by cases

I want to confirm my solution for the following problem. Problem: Let $f,g: \mathbb{R} \rightarrow \mathbb{R}$ be given by $$ f(x) = \begin{cases} 1-2x, & if & x\geq 0 \\ \left|x\right|, &...
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1answer
52 views

For an infinite sequence of functions $\Bbb{R}\to\Bbb{R}$, each function is a composition of a certain finite set of functions $\Bbb{R}\to\Bbb{R}$.

Given an infinite sequence of functions $\{g_1, g_2, \ldots, g_n, \ldots\}$ where $ g_n : \Bbb R \to \Bbb R$ prove there's a finite set of functions $ \{ f_1, f_2, \ldots, f_M \} $ such that any $ g_n ...
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0answers
43 views

Injectivity proof of multiple function composition

Given two functions $h(x)$ and $f(x)$, with $x \in X \subset \mathbb{R}^6$, with the first $3$ elements of $X$ being a unit vector. I want to prove that a map $F$ given as $F = \begin{bmatrix} h(x) \\ ...
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Doubt based on composition of functions

I have a question regarding composition of functions, which goes as follows: Let two functions are defined as $$g(x)=\begin{cases} x^2, &-1\leq x<2 \\ x+2, & 2\leq x\leq 3\\ \end{cases}$$ ...
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1answer
17 views

Graph of this function

Let f is a real valued function defined from R to R such that f(x)+f(-x)=5 Is this function even , can we plot this function ‘f’ on graph? And what information do we get from this functional equation?...
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1answer
34 views

if $f(x)=x^2+x+2$ & $g(x)=x^2-x+2$, How to prove that there is no fuction $ h:\mathbb R \to \mathbb R $ exist such as $h(f)+h(g) = g(f)$?

I tried to substitute $f(x)$ & $g(x)$ in their places but didn't find a relation; The function beginning bijective or surjective etc have nothing to do with our case I believe $g(f(x))=x^4+2x^3+...
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2answers
41 views

Find linear transformations $U,T:\textbf{F}^{2}\to\textbf{F}^{2}$ such that $UT = T_{0}$ (the zero transformation), but $TU\neq T_{0}$.

Find linear transformations $U,T:\textbf{F}^{2}\to\textbf{F}^{2}$ such that $UT = T_{0}$ (the zero transformation), but $TU\neq T_{0}$. My solution Let us consider $T(x,y) = (x,0)$ and $U(x,y) = (y,...
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1answer
19 views

Demonstrate the truthfulness of the statement

I have the following statement: Determine if is true that if $g: \mathbb{R} \to \mathbb{R}, f: \mathbb{R} \to \mathbb{R}$ and $ (g\circ f)(x) = x$ therefore $g = f^{-1}$ My attempt was: $i)$ ...
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1answer
23 views

Functional equation involving composition and exponents

Do there exist functions $f,g : R → R$ such that $f (g(x)) = x^2$ and $g( f (x)) = x^3 \text{ , }\forall x ∈ R$. Simply applying $g$ on both sides of equation $1$ and $f$ on equation $2$ ...

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