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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1answer
39 views

Functions - Algebra [on hold]

Two functions are defined by: $f(x) = 3x + 2$ $g(x) = x^2 - 4$ Find: (i) $fg(2)$ (ii) $gf(2)$ (iii) $fg(x)$ (iv) $gf(x)$ (v) the values of $x$ for which $fg(x)=17$
0
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0answers
22 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
15 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
31 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
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1answer
24 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
170 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
62 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.
-4
votes
1answer
36 views

How can I evaluate these functions [closed]

1) Let $f(x) = 3x+2$, $g(x)= \sqrt{x+1}$ . Find $f \circ g (3)$ and $g \circ f(-1)$ 2) show that $f\circ f^{-1}(x) = x$ if $f(x) = 5x+7$ 3) if $f\circ g(x) = 20 - 3/2 x$ and $f(x) = 3x+2$, find ...
0
votes
2answers
45 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
14 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. ...
0
votes
1answer
16 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
14 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
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 ...
2
votes
0answers
73 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
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
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
61 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
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 ...
3
votes
0answers
37 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
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 $f(f(f(f(x))))=0$. ...
1
vote
1answer
81 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
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 ...
1
vote
3answers
49 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
38 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
34 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
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0answers
28 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
31 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
44 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
62 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
61 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
99 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
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 ...
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
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.
6
votes
1answer
137 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 ...
15
votes
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
votes
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?
0
votes
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 ...
0
votes
0answers
21 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 ...
4
votes
1answer
26 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. ...
1
vote
0answers
28 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
votes
1answer
53 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 ...
1
vote
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 ...
0
votes
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
votes
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?