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Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

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0answers
29 views

If $f(x) =1/(1-x^2)$ and $g(x) =x$ what is $f\circ g(x)$? [on hold]

If $f(x) =1/(1-x^2)$ and $g(x) =x$ what is $f\circ g(x)$? options are A. $\sin^2 \theta $ B. $\cos^2 \theta $ C. $\tan^2 \theta$ D. $\tan(1/(1+x^2))$
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0answers
22 views

Prove that $(\mathbf g\circ\mathbf f)_*=\mathbf g_*\circ\mathbf f_*$ and $(\mathbf g\circ\mathbf f)^*=\mathbf g^*\circ\mathbf f^*$

Let $\mathbf f:\mathbf R^n\rightarrow\mathbf R^m$ and $\mathbf g:\mathbf R^m\rightarrow\mathbf R^k$. I figured out the pull-back part by finding $$ (\mathbf g\circ\mathbf f)^*(du_1)=d(g_1(f_1(x_1,...,...
-1
votes
2answers
36 views

Range of $f(x)$ when $e^x + e^{f(x)} = e$. [on hold]

How to find range of the function $f(x)$ in the equation below : $$ e^x + e^{f(x)} = e $$
0
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0answers
19 views

Find the range of the following equation

Find the range of function $$f: [0,1]\to\mathbb{R}, f(x) = x^3-x^2+4x+2\sin^{-1}x$$ I like to know calculus solution if possible.
0
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1answer
37 views

How to show $(T \circ S) \circ R=T \circ (S \circ R)$?

I am a bit new to computer mathematics, and therefore struggle a bit to understand the process of proving the equality in relation sets like this. I can both see and understand that it is correct, but ...
0
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0answers
91 views

When is $f^{p-1}(1)\equiv 1 \bmod f^{\log_mp}(1)$?

If m is coprime to p ( p prime) and $f(x)=x\cdot m$ this is the functional equivalent of Fermat's little theorem. When else is this true ? I know 1 is the multiplicative identity. Okay, so f(x)=1 is ...
4
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0answers
41 views

Every function of arity $n$ could be written as the composition of binary, unary and nullary functions? [duplicate]

A few months ago, I took some course in Logic and one of the topics was the arity of functions. The arity of a function $f$ is the number of arguments that $f$ needs to works. For example, the ...
1
vote
1answer
13 views

Interpretation of composition operator when applying a function to the output of another function

Refreshing my calculus skills a bit, I reviewed the chain rule: I wondered if the composition operation $\circ$ in $g \circ f(x)$ could actually also be written as $g(f(x))$ as this would resemble ...
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2answers
25 views

Number of compositions [closed]

Could anyone help me? I have an exercise. The number of all m-parts compositions of the number $n$ is denoted by $c (m, n)$ Prove that $c(m,n) = \binom{n-1}{m-1}$ Please explain step by step ...
0
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2answers
55 views

Bijective proof check

I have been given the problem: "Say $$f:X→Y$$ and $$g:Y→Z$$are functions so that $$g◦f:X→Z$$ is bijective and $f$ is surjective. Can you deduce that $g$ is bijective? My proof went as follows, I am ...
6
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2answers
437 views

When is composition of meromorphic functions meromorphic

When I compose a meromorphic and a holomorphic function, I get a meromorphic function. Are there other cases when a composition of two meromorphic functions is meromorphic? For example, if I compose a ...
2
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1answer
13 views

On existence of composition of functions

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be two functions. Then $g \circ f$ is clearly defined $\forall a \in A$ but what about $f \circ g$, do we take it as undefined, given that A and C are ...
2
votes
1answer
151 views

Modular arithmetic rules, of iteration of a polynomial function are?

What are the modular arithmetic properties of iterating a polynomial function ? Iteration if you aren't familiar, is repeated composition of a function with itself. It follows the rules:$$\begin{...
0
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3answers
33 views

Composing two of three given functions to obtain $2^{x+1} - 1$

A problem I'm facing involves three given functions. I need to compose two of them to obtain $2^{x+1}-1$. The three functions I'm allowed to use are $f(x) = 2x-1$ $g(x) = 1/x$ $h(x) = 2^x$ It would ...
0
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1answer
29 views

Primitive of a composite function

I'm reading Zorich, Mathematical Analysis I, and I found a not clear step in the paragraph on Primitives. The particular sentence is shown below (adapted). From the definition of primitive of a ...
1
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2answers
34 views

Identity from repeated function composition

Do functions exist such that $f^n(x)=x$ for values $n > 2$? For $n=2$ we have $-x$ and $1/x$, and for $n=3$ we can show $1/(1-x)$ is a solution. I assume that $n$ must be prime and preferably ...
0
votes
1answer
37 views

commutativity of rotations and reflections

The question is as follows: [Concerning the square embedded in the plane,] prove that the $90^{\circ}$ clockwise rotation $\sigma$ and the reflection through the north/south axis $\rho$ do not ...
0
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1answer
33 views

Prove that a relation R on A is antisymmetric if and only if $R∘R^{-1} \subseteq I_A$

If: I tried to take some element $(a,b)∈R∘R^{-1}$ and show that we have some $c∈A$ s.t. $(a,c)∈R^{-1}$ and $(c,b)∈R$, and use the fact that $R∘R^{-1} \subseteq I_A$ which gives that $a=b$. I then ...
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0answers
40 views

Can any function be rewritten as a composition of functions on $x$?

Many functions use forms that have multiple instances of $x$, including polynomials and rational functions. However, finding the domain and range can be simpler when they are written with only one use ...
0
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2answers
26 views

How do you determine the domain of a composition of functions?

Question: How would you determine the domain of g(f(x)) with g(x) = 3/x and f(x) = 6/(1-4x)? Attempt: If you composite the function, the final answer would be (1-4x)/2. However, the domain would not ...
0
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1answer
29 views

Help on composition of functions

I know how to compose normal functions but never seen this type of task any help? Find the composition $g \circ f$ of the following functions $f, g : \Bbb{R} \to \Bbb{R}$ given by the formulas: $$...
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0answers
25 views

Is there a formal connection between transitivity and compositionality?

Transitivity is the property of a relation: $R_{ab}\land R_{bc} \to R_{ac}$. Compositionality is an operation on functions: $\circ : (A\to B) \times (B \to C) \to (A \to C)$. Intuitively, these are ...
3
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1answer
42 views

$f(x)$ from $f(g(x))$

Is it always possible to find $f(x)$ if the composite function $h(x) = f(g(x))$ and $g(x)$ are given? In other words, can there be any cases where, for given $h(x)$, we can not express it in an ...
1
vote
2answers
34 views

Composition of discontinuous functions

Let $f(x) = [x]$ and $$ g(x)=\begin{cases} 0&\text{if}\;x \in \Bbb Z\\x^2&\text{otherwise}\end{cases}$$ Is $g\circ f$ continuous? I know conditions of continuity but in case of composition ...
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votes
1answer
79 views

If $f(x) = \dfrac{x - 2011}{11}$, then evaluate $(f \circ f \circ f \circ f \circ f)(x)$

If $f(x) = \dfrac{x - 2011}{11}$, then $(f \circ f \circ f \circ f \circ f)(x)$ is $\cdots$ A. $\frac{x+2011}{x-1}$ B. $\frac{x+2011}{x+1}$ C. $\frac{x-2011}{x+1}$ D. $\frac{x-2011}{x-1}$ E. $\frac{-...
0
votes
1answer
29 views

Evaluating the $k$th derivative of composite $g\circ f$ if $f^{\prime}(0) = 0$ without using Bruno's theorem

Suppose $f$ and $g$ are infinitely differentiable. Suppose $f^{\prime}(0) = 0$. Consider the composite function $g \circ f$. If I wanted to calculate the $k$th derivative of $g \circ f$ and evaluate ...
4
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1answer
120 views

How to show that $\dim\ker(AB) \le \dim \ker A + \dim \ker B $?

I want to show that $$ \dim \ker(AB) \le \dim \ker A + \dim \ker B. $$ My problem I thought that I can do that in this way: Let consider $x \in\ker B$ $$Bx = 0$$ Let multiplicate this from left ...
14
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2answers
159 views

Are there two functions $f, g$ such that $f(g(x)) = x^3$ and $g(f(x)) = x^5$?

Question. Are there two functions $f, g: \mathbb{R}\rightarrow\mathbb{R}$ that satisfy $f(g(x)) = x^3 \enspace\forall x\in\mathbb{R}$ and $g(f(x)) = x^5\enspace\forall x\in\mathbb{R}$? This is an ...
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3answers
72 views

Why is the composition of a surjective and injective function neither surjective nor injective? [closed]

I am currently preparing for an exam coming up and I was therefore looking in earlier exam-sets, and found a question with a solution I just cannot make sense in my head. Let $A, B, C$ be three ...
4
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2answers
126 views

$f(x)$ is continuous at $x=\alpha$ ,$g(x)$ is discontinuous at $x=a$ but $g(f(x))$ is continuous at $x=\alpha$

Suppose $f,g:\mathbb{R} \rightarrow \mathbb{R}$ are such that $f(x)$ is continuous at $x=\alpha$ and $f(\alpha)=a$ and $g(x)$ is discontinuous at $x=a$, but $g\big(f(x)\big)$ is continuous at $x=\...
4
votes
1answer
69 views

Iterated function 'periodicity'

Note: $f^n$ denotes the iteration of composition, e.g. $f^3(x)=(f\circ f\circ f)(x)$ I've noticed that particular functions have a certain property where for some number $n$ the iterations of the ...
1
vote
1answer
17 views

Domain of a composite function without having g(x) function

Suppose that the domain of "$f$" function is equal to $[0,1]$, then find the domain of: $f(3x^2)$ $f(1-x)$ and $f(sin x)$ I know that $f(3x^2)$ is the same that we say $g = 3x^2$ so it's domain is $R$ ...
1
vote
1answer
25 views

Canonical factorization of a certain function

Given the sets $A=${$x_i|1\leq i\leq 9$} and $B=${$y_j|1\leq j\leq 6$} we consider the map $f:A\to B$ defined by: $$f(x_1)=y_1 \qquad f(x_2)=y_1 \qquad f(x_3)=y_3 \\ f(x_4)=y_3 \qquad f(x_5)=y_3 \...
3
votes
2answers
85 views

What is the $n$-time iterated adjugate of an $n\times n$ matrix $A$?

What is $\underbrace{\text{adj}\Big(\text{adj}\big(\ldots(\text{adj}}_{n\text{ adj}}\ A)\ldots\big)\Big)$, where $\text{adj}$ is written $n$ times, and the order of the matrix $A$ is $n\times n$? Can ...
7
votes
2answers
90 views

Find function $f(x)$ that satisfying differential relation

Suppose the functions $F(x)$ and $G(x)$ satisfying $$F(x)=f(x)-\frac{1}{f(x)}$$ $$G(x)=f(x)+\frac{1}{f(x)}$$ such that $F'(x)=(G\circ G)(x)$, with initial condition $f(\frac{\pi}{4})=1$ is given....
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0answers
18 views

Derivative of an infinite composition of functions

Let $f(x)=g(g(g(...g(x))))$, where the function $g$ is applied to $x$ and infinite amount of times. I am assuming that $x$ is real. What is special about points at which $df/dx=0$? Are they related to ...
3
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1answer
26 views

Limit of the composistion of two functions when the limit of $f$ does not exist?

The theorem about the limit of composition of two real functions $f$ and $g$ is proved here. But it is required that the two limits (of $f$ and $g$) both exist. I can't understand how to deal with ...
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3answers
39 views

If $(f\circ g)(x)=\tan^2x$ and $g(x)=\sqrt{\cos2x}$ then find $f(x)=?$

We've given : $$(f\circ g)(x)=\tan^2x$$ and $$g(x)=\sqrt{\cos 2x}$$ Then how to find the function $f(x)$? I know that $$(f\circ g)(x)=f(g(x))= f( \sqrt{\cos2x})$$ But I do not know how to find $f(x)$...
2
votes
1answer
35 views

Can composition of morphisms in a category be carried out on any subgraph of a commutative diagram, in-place?

Here is what the rule looks like to us and how we specify it to the app I'm writing. I was wondering can you take any commutative diagram $J$ and apply this rule to a subgraph matching $A \...
0
votes
0answers
31 views

What is the n-fold composition of a linear map “T”, if n is zero?

I am doing an assignment on Groups of Linear Transformations. Here is a defnition with regards to n-fold compositions: "Let $T \in \mathcal{L}(V)$ denote the n-fold compositions of $T$ with itself as ...
0
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0answers
29 views

How do I use the Composite of Continuous Functions theorem to show that a function is continuous?

$$f(x)= \sqrt \frac{x}{x+1}$$ I know that I have to split it into two separate functions: $$ g(x)= \sqrt x$$ $$h(x)=\frac{x}{x+1}$$ I'm just not sure what to do next, and I can't seem to find anything ...
0
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1answer
54 views

How do describe the composition of an indicator function?

I have an indicator function: I have to describe the composition in this form: I replaced the 'n' value with the function but I cannot simplify it, how can one describe and simplify the composition ...
0
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1answer
19 views

Proposition involving inner product and derivatives

I've found this proposition and I don't know how to prove it. The proposition is the following: Let's have $f:U\subset E\rightarrow R^{n}$, $E$ normed, $U$ open in $E$. Let's pick $p,q \in U, [p,q] \...
0
votes
2answers
74 views

Number of zeros of compositions of $f(x) = 4x(1-x)$

We have the function $f: [0,1] \rightarrow [0,1]$ such that $f(x) = 4x(1-x)$. Find the number of zeros of $$f^{\circ n}(x)=f(f(...f(x)))...)$$ ($n$ times composition of $f()$). I think that the answer ...
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votes
1answer
18 views

Is this an unsolvable function composition problem? Or an absurd?

I am solving a PDE problem. In a certain point, I achieve the following equality: $G(3x+1) = y^2 + e^x$ This is,basically, a problem involving function composition. How do I find $G(x)$? I have ...
0
votes
3answers
49 views

Given 2 functions $f,g:\mathbb{R} \to \mathbb{R}$ with $g \circ f$ surjective. Show that $f$ does not need to be surjective.

A friend has given me this task and asked for help. I immediately tried picking various $f$ and $g$ such that $g$ and $g\circ f$ are both surjective but $f$ is not. However, i always found myself ...
0
votes
1answer
31 views

Function composition according to relation

Let $f: A \rightarrow A \ $ and a relation $R_f$ over $A$ $\, $s.t.: $R_f = \left\{(x,y) \in A \times A \mid y=f(x)\right\}$ Show if the following statements are true or false and explain your choice: ...
0
votes
1answer
79 views

What is the domain and range of this relation in?

From: Triamudom Add.math sheet pg.19 This is the progress I’ve done so far. As you can see I couldn’t find a way to arrange $x$ in terms of $y $ or vice versa which is required to solve for the ...
4
votes
4answers
50 views

If limit of $f$ is $L$ and limit of $g$ is $M$, then limit of $g$ composed $f$ is $M$?

Problem: Find examples of functions $f$ and $g$ defined on $\mathbb{R}$ with $\lim\limits_{x\to a}f(x) = L$, $\lim\limits_{y\to L}g(y) = M$, and $\lim\limits_{x\to a} g(f(x))\neq M$. I have tried ...
2
votes
0answers
53 views

Solving equations with composed constrained functions

I was lately curious about an iterative approach that would solve maths equations containing composed functions with contraints. For example, if I have the following equation: $$ f(g(h(w))) = 0 \...