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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73 views

Inverse of multivariated polynomials over finite fields [on hold]

I was thinking about if there is a general method, or at least case-by-case method, of expressing inverse functional compositions on Galois Fields. For instance, in $GF(2)$ how to express the inverses ...
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1answer
13 views

Composition of 2 monotonic functions

Let $f$ be a monotonic function $f:[a,b] \rightarrow\mathbb{R}$ and $g$ be a monotonic function $g:[c,d]\rightarrow[a,b]$. Show that $f\circ g$ is monotonic
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2answers
11 views

Given two piece-wise functions, prove their composition is periodic.

I know how to graph h(f(x)), however, I am having trouble proving it is periodic. i. We must prove that h(f(x + k)) = h(f(x)) for all x ∈ ℝ and for some k ≠0 ∈ ℝ in order to prove that h(f(x)) is ...
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2answers
29 views

$f \circ f^{-1} = i_B$ proof using the fact that $f^{-1} \circ f = i_A$

Suppose f is function from A to B, and suppose that $f^{-1}$ is a function from B to A. Assume $f^{-1} \circ f = i_A$. Then show therefore that $f \circ f^{-1} = i_B$. I tried applying left ...
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72 views

We all know about compositions of functions, but what about decomposition. Is there a way with math, not just heuristics?

The composition operator is a well know and quite often used method in integration and differentiation, think u-substitution. However, given a composition like $$f(f(f(...f(x)...)))$$ Where there are ...
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0answers
27 views

functions compositions, three sets, counting compositions

Given three sets $P, Q, R$ such that $|P|=p, |Q|=q, |R|=r,$ and $p,q,r > 1$ let $f(x): P\rightarrow Q$, and $g(x):Q\rightarrow R$ be two functions. Find the number of functions which can be ...
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13 views

Internalizing results about composition and surjectivity/injectivity

I'm trying to see if there is any intuition pump / analogy that allows me to internalize ( and readily derive them) a series of results about the concepts of composition mixed with ...
2
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1answer
56 views

How to find $s(\exp(d(x)))$ ~ $ x + 2 $?

Let $x$ be a positive real. I want to find a pair of analytic functions $s(x),d(x)$ such that $s(d(x)) = x$ and $ s(\exp(d(x)))$ ~ $ x + 2 $ More presicely I Also want : $$ \lim_{x \to \infty} ...
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1answer
59 views

Geometric description of composition of two convex polygons? [duplicate]

The composition was defined as follow: $(a,b)$ $\in$ $(R;S)$ $\Leftrightarrow$ $\exists c | (a,c) \in R \\ $&$ (c,b) \in S$ . Also the composition was defined as the binary product of two ...
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23 views

Writing a set in terms of compositions of known functions.

for this problem we are to write a the given set using a composition of known functions like map, dist, seq, pairs, ., +, -, etc. The given set is: $\langle n, n - 1, n - 2, ..., 0 \rangle$. I am ...
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36 views

Which symbol to use for composition of a sequence of functions [duplicate]

I know how to write the composition of two functions: $f\circ g$ but I don't know whether there's a standard symbol for a sequence (similar to $\sum_i{f_i}$, $\prod_i{f_i}$ or $\bigotimes_i{f_i}$, ...
3
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1answer
55 views

If $ f(x) = \left\{\frac{3x}{2}\right\}\;,$ Where $\{x\}=x-\lfloor x \rfloor$. Then real solution of $f(f(x))=0$

If $\displaystyle f(x) = \left\{\frac{3x}{2}\right\}\;,$ Where $\{x\}=x-\lfloor x \rfloor$. Then no. of solution of the equation $f(f(x))=0$ and $f(f(f(x)))=0$ and $f(f(f(f(x))))=0$. ...
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1answer
77 views

What is a composition of two binary relations geometrically?

the composition was defined as follow: (a,b) \in (R;S) <=> there is c | (a,c) \in R and (c,b) \in S . If our two relations R and S are two convex polygon ...
3
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1answer
74 views

When does differentiability of $g\circ f$ and $f$ resp. $g$ imply differentiablity of $g$ resp. $f$?

To me the following seems intuitively true: If $f$ is differentiable at $x$ with surjective derivative then $g$ is differentiable at $f(x)$ iff $g\circ f$ is differentiable at $x$. On the other ...
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3answers
47 views

Can the composition of two non-invertible functions be invertible?

(Context: I came across this exercise in the textbook "Coding the Matrix" when reading it to supplement my studies in the Coursera class "Coding the Matrix".) After proving that the composition of ...
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1answer
37 views

Continuity of a nonlinear operator on fractional-order Sobolev spaces

Let $N\colon \mathrm{H}^s(\mathbb{R}) \to (\mathrm{H}^s(\mathbb{R}))^*$, where $s > \frac{1}{2}$, be an operator given by $N(u) = \langle u^p, \cdot \rangle_{\mathrm{L}^2(\mathbb{R})}$ for a fixed ...
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1answer
27 views

Is transitive Relation closed under composition?

it's true that equivalence relations is closed under composition, i.e., if R is a equivalence relation RoR is so.(Because RoR =R) But this not imply that any transitive Relation is so. Briefly; i ...
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0answers
26 views

How do I show that infinite application of this function gives a constant?

I want to show that g(x) returns the same value independent of x and hence is a constant. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
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1answer
29 views

Can a vector function be considered a composite function?

I'm just curious about this. Technically, can a vector function be considered a composite function? Are they equivalent? For example, determining the domain of a vector function, will it be the same ...
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2answers
42 views

Domain and Range for composite function

Given the function $f(x) = x^2$ with the domain $[0, \infty)$ and $g(x) = \sin(x)$ with domain $(- \infty, \infty)$. What are the domain and range of $f(g(x))$ and $(g(f(x))$? I start the ...
2
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1answer
55 views

Iterated function

Let $$f(x)=x−\frac{1}{x}$$ Find the number of real solutions to $f(f(f(f(x))))=1$. Do I evaluate it completely, or is there some other way. After third composition it got nasty, so I left it.
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2answers
53 views

Really confused about one-one,onto and invertibility.

I am really have some difficulty understanding how to do this problem. It asks to show that if T is one-to-one and onto, then T is invertible, and why T being invertible is equivalent to being one to ...
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5answers
81 views

How do I find the kernel of a composition of functions?

Functions $g$ and $f$ are linear and injective. How do I go about finding the kernel of $g \circ f$? I'm asking because I want to prove that $\ker(f) = \ker(g \circ f)$.
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2answers
35 views

Given $f(x)=x+2$ and $g(f(x))=3x^2+12x+5$. find g(x) [closed]

Please help, its for a maths investigation so my teacher cant help me. Thanks!!
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1answer
17 views

Proving $\forall f\in R^{ R}\left [(f(6)=6) \to (\exists g\in R^{ R}((g\neq i_{\mathbb R})\wedge (g\neq f)\wedge (f\circ g = g\circ f)) \right ]$

Prove $\forall f\in \mathbb R^{\mathbb R}\left [(f(6)=6) \to (\exists g\in \mathbb R^{\mathbb R}((g\neq i_{\mathbb R})\wedge (g\neq f)\wedge (f\circ g = g\circ f)) \right ]$ My attempt: Let ...
0
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2answers
22 views

For a composition to be defined: $Domf\circ g\subseteq Dom f, Im f\circ g \subseteq Im g $?

For a composition to be defined, is the following two a must? $$f:A\to B, g: C\to D\\ f\circ g : C\to B \\ Domf\circ g\subseteq Dom f\\ Im f\circ g \subseteq Im g $$ Are there other conditionals for ...
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2answers
17 views

Order of composition when dealing with transformations

I have been struggling with a question in my book. $T$ is a translation of $(+5,+4)$, $M$ is a reflection in the line $y=x$. $R$ is a 90 degree anticlockwise rotation about $(0,0)$ Write down ...
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6 views

Finding functions such that $F\circ g_k=i_{\mathcal P (\mathbb R)}$

Let $F:(\mathbb R\times \mathcal P (\mathbb R))\to \mathcal P (\mathbb R) \\ F((x,A))=\{y\in \mathbb R| \frac {x+y} 2\in A \}$ Define two different functions $g_k:\mathcal P (\mathbb R)\to ...
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1answer
26 views

Calculate the limit of a composite function

$\lim \limits_{x \to -\infty} \log(\cos\frac1x)(x^3-3x+\sin x)$ Is L'Hôpital's rule a way to evaluate this limit? Any suggestions would be appreciated.
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1answer
136 views

Can $f(g(x))$ be a polynomial?

Let $f(x)$ and $g(x)$ be nonpolynomial real-entire functions. Is it possible that $f(g(x))$ is equal to a polynomial ? edit Some comments : I was thinking about iterations. So for instance ...
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4answers
295 views

Find $f''(x)$ if $f\circ f'(x) = 4x^2 + 3$

Can you tell me the solution of this question? If: $f\circ f'(x)=4 x^2 +3$ then what is $f''(x)$? This was a question in math test which I just took yesterday. One function satisfying the ...
2
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3answers
42 views

If $f\circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?

If $f\circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?
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3answers
39 views

Decompose a polynomial: find $f(x)$ such that $h(x) = f(g(x))$

I try to make an algorithm that decomposes a polynomial, ie find $f(x)$ such that $h(x) = f(g(x))$ by knowing $h$ and $g$. For example, having : $h(x) = 112x^6 + 1232x^5 + 2772x^4 - 3388x^3 + 847x^2 ...
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0answers
17 views

What is the domain and image of the composition of mappers in a manifold

I was trying to understand the following: which I got from: http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf I was wondering, why the domain was: $$ \alpha(U_{\alpha} \cap ...
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1answer
25 views

Showing that f,g are invertible if $A$ is a finite set and $f,g: A\to A$ such that $f\circ g$ is invertible

Let $A$ be a finite set and $f,g: A\to A$ such that $f\circ g$ is invertible. Prove f,g are invertible. Prove that if $A$ is an infinite set, it doesn't mean that f,g are invertible. ...
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0answers
26 views

Finding out if a function is invertible: $f,g:\mathbb N\to \mathbb N$, $g(x)=2x$ and $f$ with cases

Let $f,g:\mathbb N\to \mathbb N$ such that $g(x)=2x$ and $f(x)=\begin{cases}\frac x 2 &, x\in\mathbb N_{even}\\ x+9 &,x\in\mathbb N_{odd}\end{cases}$ ...
0
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1answer
51 views

Can every function be a composite to itself and how to know if a composite between two functions is defined?

Can every function be a composite to itself? like we have $f:A\to B$ is $f \circ f$ always defined? Can we say that if $f$ is a injection/surjection/bijection then so is $f\circ f$? Also, how do ...
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1answer
17 views

If $S = \{ (p, q) \in P \times P \mid \text{p is an ancestor of q}\}$, what is $S \circ S^{-1}$?

Suppose $S = \{ (p, q) \in P \times P \mid \text{p is an ancestor of q}\}$. What is $S \circ S^{-1}$? To achieve the desired result, I would start by identifying what $S^{-1}$ (the inverse of ...
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2answers
18 views

If $g\circ f$ is $1$-$1$ then $f$ is $1$-$1$ but $g$ is not necessarily $1$-$1$.

Let $f:X\longrightarrow Y$ and $g: Y\longrightarrow Z$. Show that, if $g\circ f$ is $1$-$1$, then $f$ is $1$-$1$, but $g$ is not necessarily $1$-$1$ I don't know how to start the proof. We have ...
0
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1answer
30 views

Composition of relations - check my method

I just want to check that the method I am using for the composition of relations is right. If a pair in R (z,y) and a pair in S (x,z) then (x,y) yield and become a pair in SoR?
3
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1answer
44 views

Is there a multiple function composition operator?

Is there a commonly-accepted operator which defines multiple function composition? I have not been able to find one on any of the related Wikipedia pages. In one of my proofs, I've been finding it ...
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0answers
26 views

Regarding functions composition in descrete math

Let $f : A \to B$ be a function. Let $g:B \to B$ and $ℎ:B \to B$ be functions that are total surjective and injective (bijection). Prove that if $h \circ g \circ f$ is total and injective, then $f$ is ...
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2answers
21 views

Domain and Range of Function Composition

Given: $a\left(x\right)=e^x$ $b\left(x\right)=\left|x+2\right|$ $c\left(x\right)=\frac{\left(x-2\right)}{\left(x+1\right)}$ What is: $\left(\frac{a\cdot ...
0
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1answer
44 views

How to check convexity of a composition when some properties of inner and outer functions are known?

If $g(x)$ function is concave in $x$, and we want $g( f(x) )$ (where $f(x)$ is another function) to be convex in $x$, what are the required properties of $g(x)$ and $f(x)$? It would be appreciated ...
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1answer
33 views

Composition of Invertible Functions

Once again we're studying domain and range in class and I encountered this problem. If $f(x)$ and $g(x)$ are both invertible functions, and the domain and range of each function is the set of real ...
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2answers
39 views

Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ ...
0
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3answers
57 views

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true? (a). If $g$ is continuous, then $f\circ g$ is continuous. ...
0
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0answers
47 views

Is this proof show that the derivative of zeta function has no zeros in the critical strip?

suppose that :( $k \circ \zeta )(s)$= $(\zeta \circ k )(s) \neq 0 $.....$(1)$ ,and suppose that $ k(s)=\zeta(1-s)$ where : $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. and ...
1
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1answer
33 views

Is the composition of a set of convex functions convex?

Here we see the proof for $f(x)$ being convex where $$f(x) = h(g(x))$$given $h$ is convex and nondecreasing and $g$ is convex. But what if $$f(x) = h(g_1(x),g_2(x),g_3(x),...,g_k(x))$$ where each ...
0
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0answers
28 views

Fourier Transform of a function with sinusoidal sampling

What is the relation between the Fourier Transform (FT) of $f(x)$ with regular sampling and the FT of $f(x)$ with sinusoidal sampling? In other words, it's a FT of a function composition $f\circ ...