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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11
votes
7answers
289 views

Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) ...
-5
votes
0answers
28 views

Composition of Relations solving p 0 σ and σ 0 p [on hold]

Explain me difference between p 0 σ and σ 0 p and how to get the answer.
0
votes
0answers
19 views

Are these two compositions of two functions differentiable?

Assuming $U=\{x\in\mathbb{R}^2:x_1^2+x_2^2<1\}$ is the open unit circle in the plane and $f,g:U\rightarrow\mathbb{R}^2$ two functions with $f(0)=g(0)=0$. $f$ is Fréchet-differentiable in $0$, and ...
2
votes
1answer
28 views

Finding a (nonidentity) rational map of the plane with period $7$

Does there exist a nonidentity (which also is not a rotation) rational map $f:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ with period $7$, i.e., for which the seventh iteration $f^7$ is the identity ...
0
votes
0answers
13 views

Composition of Lp convergent function and continuous function

Let $f$ be a continuous function on $\mathbb R$ such that for $U\subset \mathbb R ^n$ bounded it holds that $\forall w\in L^p(U) ~~ f(w)\in L^q(U)$. Let $~u_k \rightarrow u$ in $L^p(U)$ . Does ...
1
vote
0answers
20 views

Convergence in probability of function composition.

I need to show that $G_n \stackrel{P}{\to}_n F_0$, i.e. for any $\epsilon>0$ $$ P(|| G_n - F_0||>\epsilon) \to_n 0 $$ We know the following: $G_n$ and $F_0$ are a bilinear functions from ...
1
vote
1answer
15 views

While finding points of discontinuity for a composite function do I need to consider the points of discontinuity of individual functions too.

I'm solving problems based on composition of functions and stuck in this problem. If $f(x)=\frac{1}{(x-1)(x-2)}$ and $g(x)=\frac{1}{x^2}$, then find the points of discontinuity of $f(g(x))$. We ...
1
vote
1answer
26 views

Struggling with a problem in functions.

Suppose '$f$' is a continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $f(f(a))=a$ for some $a \in \mathbb{R}$ then find the number of solutions of the equation $f(x)=x$. Options given: ...
0
votes
0answers
10 views

Composition of functions class $C^n$

Suppose I have 2 functions class $C^n$ and I consider their composition. Would that still be a $C^n$ function? If so why (I demand a proof). It seems logical to me that it is true but I can't find a ...
0
votes
3answers
76 views

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective.

For a function Y : X→X , if Y is injective, then Y∘Y∘Y is injective. My attempt: Using contrapositive, if Y is not injective. then Y ∘ Y is not injective, the there exist x, x' ∈ X with x ≠ x' but ...
2
votes
1answer
34 views

Composing Linear Transformations

Hello and thank you in advance; The problem: "Let V be a vector space and T a linear operator $T:V\rightarrow V $, show that $$[T^m]_B =[T]_B^m$$ Where $B$ is a basis(any) of $V$ and $T^m=T\circ T ...
6
votes
1answer
50 views

which functions can be obtained as a composition of a continuous function with itself? [duplicate]

let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for ...
-2
votes
1answer
26 views

How to find $g(x)$, if: $f(x)=(x+1)/x$, and $f(g(x))=x$?

How to find $g(x)$, if: $f(x)=\frac{(x+1)}{x}$, and $f(g(x))=x$? I know that the answer is that $g(x)=\frac{1}{(x-1)}$ But how to come to that answer remains a mystery to me Please give me some ...
0
votes
0answers
25 views

Differentiate $g\circ f$ transformation

Differentiate $g \circ f$ of the following functions: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$f(x,y)=(x-y,x+y)$$ $$g: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $$ $$g(x_1,x_2)=(e^{x_1} \cos ...
1
vote
1answer
16 views

How to determine a function from a sequence of consecutively composed functions?

Let $ f(x) = x+1 $ and $g(x) = 2x$ Prove $f^2g = gf $ and determine $f^igf^jgf^k(x)$ explicitly as a function of x and in terms of i,j,k. I got through the proof but I don't understand what the ...
2
votes
1answer
33 views

If a composition of two maps is smooth, as well as one of the maps, then so is the other.

Let $M$, $N$, and $K$ be smooth manifolds, and consider the maps $g:M\to N$, and $f:N\to L$. Assume that the composition $f\circ g$ is smooth. If any of $f$ and $g$ is smooth can we conclude that the ...
1
vote
1answer
27 views

Permutations, compositions and associativity properties

Let n be a postive integer, and let σ : {1, . . . , n} → {1, . . . , n} be a one-to-one and onto map. Then σ is called a permutation on n elements. The set of all permutations on n elements is denoted ...
5
votes
2answers
182 views

The composition of a nowhere-differentiable function with a differentiable function.

This is actually Problem $ 17 $ from Chapter $ 10 $ of the Fourth Edition of Michael Spivak’s Calculus. The statement is quite simple, but I have not had any success in finding an example. Here is the ...
0
votes
3answers
83 views

If $ f $ is injective and $ g $ is injective, then $ f \circ g $ is surjective. [duplicate]

I can prove that if $ f $ and $ g $ are both injective, then $ f \circ g $ is injective, but I don’t know how to prove that $ f \circ g $ is surjective.
0
votes
2answers
47 views

Show that $g$ is one-one if and only if $g$ is onto.

Original problem A function $g$ from a set $X$ to itself satisfies $g^m=g^n$ for positive $m$ and $n$ with $m>n$. Here $g^n$ stands for $g\circ g\circ \dots g$(n times). Show that $g$ is one-one ...
0
votes
1answer
16 views

Taylor Expansion of Composition of Functions

I have in my lecture notes that $$f(g(x+h))-f(g(x))\approx f'(g(x))\left(g(x+h)-g(x)\right)+f''(g(x))(g(x+h)-g(x))^2$$ He explained can found via taylor expansion, but I try to expand it and am not ...
0
votes
0answers
43 views

Linear transformation of eigenspace is subset of eigenspace

Let $V$ be a vector space over a field $\mathbb{F}$ and let $L$, $M$ be two linear transformations from $V$ to itself. a. Show that the subset $W= {x ∈ V : L(x) = M(x)}$ is a subspace of $V$ b. ...
1
vote
1answer
23 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
0
votes
2answers
16 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 ...
0
votes
2answers
30 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 ...
2
votes
0answers
76 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 ...
2
votes
0answers
29 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 ...
1
vote
0answers
15 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
votes
1answer
58 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} ...
1
vote
1answer
63 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 ...
0
votes
0answers
24 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 ...
3
votes
0answers
51 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
votes
1answer
56 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 $f(f(f(f(x))))=0$. ...
1
vote
1answer
89 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
votes
1answer
75 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 ...
1
vote
3answers
53 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 ...
1
vote
1answer
44 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 ...
0
votes
1answer
49 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 ...
1
vote
0answers
29 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. $$g(x) = \lim_{n \rightarrow \infty}(\underbrace{f \circ f \circ \cdots \circ f}_{n\text{ times}})(x)$$ ...
-1
votes
1answer
39 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 ...
1
vote
2answers
49 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
votes
1answer
66 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.
2
votes
2answers
69 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 ...
3
votes
5answers
127 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 $\ker(f) = \ker(g \circ f)$.
0
votes
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!!
2
votes
1answer
18 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
votes
2answers
24 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 ...
0
votes
2answers
20 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 ...
1
vote
0answers
7 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 ...
0
votes
1answer
27 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.