Questions tagged [function-and-relation-composition]
For questions about the composition of functions and relations.
-2
votes
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. $\tan(1/(1+x^2))$
0
votes
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 $$
0
votes
0answers
19 views
Find the range of the following equation
Find the range of function
$$f: [0,1]\to\mathbb{R}, f(x) = x^3-x^2+4x+2\sin^{-1}x$$
I like to know calculus solution if possible.
0
votes
1answer
37 views
How to show $(T \circ S) \circ R=T \circ (S \circ R)$?
I am a bit new to computer mathematics, and therefore struggle a bit to understand the process of proving the equality in relation sets like this. I can both see and understand that it is correct, but ...
0
votes
0answers
91 views
When is $f^{p-1}(1)\equiv 1 \bmod f^{\log_mp}(1)$?
If m is coprime to p ( p prime) and $f(x)=x\cdot m$ this is the functional equivalent of Fermat's little theorem. When else is this true ? I know 1 is the multiplicative identity. Okay, so f(x)=1 is ...
4
votes
0answers
41 views
Every function of arity $n$ could be written as the composition of binary, unary and nullary functions? [duplicate]
A few months ago, I took some course in Logic and one of the topics was the arity of functions. The arity of a function $f$ is the number of arguments that $f$ needs to works. For example, the ...
1
vote
1answer
13 views
Interpretation of composition operator when applying a function to the output of another function
Refreshing my calculus skills a bit, I reviewed the chain rule:
I wondered if the composition operation $\circ$ in $g \circ f(x)$ could actually also be written as $g(f(x))$ as this would resemble ...
-1
votes
2answers
25 views
Number of compositions [closed]
Could anyone help me? I have an exercise.
The number of all m-parts compositions of the number $n$ is denoted by $c (m, n)$
Prove that
$c(m,n) = \binom{n-1}{m-1}$
Please explain step by step
...
0
votes
2answers
55 views
Bijective proof check
I have been given the problem: "Say $$f:X→Y$$ and $$g:Y→Z$$are functions so that $$g◦f:X→Z$$ is bijective and $f$ is surjective. Can you deduce that $g$ is bijective?
My proof went as follows, I am ...
6
votes
2answers
437 views
When is composition of meromorphic functions meromorphic
When I compose a meromorphic and a holomorphic function, I get a meromorphic function. Are there other cases when a composition of two meromorphic functions is meromorphic? For example, if I compose a ...
2
votes
1answer
13 views
On existence of composition of functions
Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be two functions.
Then $g \circ f$ is clearly defined $\forall a \in A$ but what about $f \circ g$, do we take it as undefined, given that A and C are ...
2
votes
1answer
151 views
Modular arithmetic rules, of iteration of a polynomial function are?
What are the modular arithmetic properties of iterating a polynomial function ?
Iteration if you aren't familiar, is repeated composition of a function with itself. It follows the rules:$$\begin{...
0
votes
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
votes
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
vote
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
votes
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 ...
0
votes
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
votes
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
votes
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:
$$...
1
vote
0answers
25 views
Is there a formal connection between transitivity and compositionality?
Transitivity is the property of a relation: $R_{ab}\land R_{bc} \to R_{ac}$. Compositionality is an operation on functions: $\circ : (A\to B) \times (B \to C) \to (A \to C)$.
Intuitively, these are ...
3
votes
1answer
42 views
$f(x)$ from $f(g(x))$
Is it always possible to find $f(x)$ if the composite function $h(x) = f(g(x))$ and $g(x)$ are given?
In other words, can there be any cases where, for given $h(x)$, we can not express it in an ...
1
vote
2answers
34 views
Composition of discontinuous functions
Let $f(x) = [x]$ and $$ g(x)=\begin{cases} 0&\text{if}\;x \in \Bbb Z\\x^2&\text{otherwise}\end{cases}$$
Is $g\circ f$ continuous?
I know conditions of continuity but in case of composition ...
-1
votes
1answer
79 views
If $f(x) = \dfrac{x - 2011}{11}$, then evaluate $(f \circ f \circ f \circ f \circ f)(x)$
If $f(x) = \dfrac{x - 2011}{11}$, then $(f \circ f \circ f \circ f \circ f)(x)$ is $\cdots$
A. $\frac{x+2011}{x-1}$
B. $\frac{x+2011}{x+1}$
C. $\frac{x-2011}{x+1}$
D. $\frac{x-2011}{x-1}$
E. $\frac{-...
0
votes
1answer
29 views
Evaluating the $k$th derivative of composite $g\circ f$ if $f^{\prime}(0) = 0$ without using Bruno's theorem
Suppose $f$ and $g$ are infinitely differentiable. Suppose $f^{\prime}(0) = 0$. Consider the composite function $g \circ f$. If I wanted to calculate the $k$th derivative of $g \circ f$ and evaluate ...
4
votes
1answer
120 views
How to show that $\dim\ker(AB) \le \dim \ker A + \dim \ker B $?
I want to show that
$$ \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$$
Let multiplicate this from left ...
14
votes
2answers
159 views
Are there two functions $f, g$ such that $f(g(x)) = x^3$ and $g(f(x)) = x^5$?
Question. Are there two functions $f, g: \mathbb{R}\rightarrow\mathbb{R}$ that satisfy $f(g(x)) = x^3 \enspace\forall x\in\mathbb{R}$ and $g(f(x)) = x^5\enspace\forall x\in\mathbb{R}$?
This is an ...
-1
votes
3answers
72 views
Why is the composition of a surjective and injective function neither surjective nor injective? [closed]
I am currently preparing for an exam coming up and I was therefore looking in earlier exam-sets, and found a question with a solution I just cannot make sense in my head.
Let $A, B, C$ be three ...
4
votes
2answers
126 views
$f(x)$ is continuous at $x=\alpha$ ,$g(x)$ is discontinuous at $x=a$ but $g(f(x))$ is continuous at $x=\alpha$
Suppose $f,g:\mathbb{R} \rightarrow \mathbb{R}$ are such that $f(x)$ is continuous at $x=\alpha$ and $f(\alpha)=a$ and $g(x)$ is discontinuous at $x=a$, but $g\big(f(x)\big)$ is continuous at $x=\...
4
votes
1answer
69 views
Iterated function 'periodicity'
Note: $f^n$ denotes the iteration of composition, e.g. $f^3(x)=(f\circ f\circ f)(x)$
I've noticed that particular functions have a certain property where for some number $n$ the iterations of the ...
1
vote
1answer
17 views
Domain of a composite function without having g(x) function
Suppose that the domain of "$f$" function is equal to $[0,1]$, then find the domain of:
$f(3x^2)$ $f(1-x)$ and $f(sin x)$
I know that $f(3x^2)$ is the same that we say $g = 3x^2$ so it's domain is $R$
...
1
vote
1answer
25 views
Canonical factorization of a certain function
Given the sets $A=${$x_i|1\leq i\leq 9$} and $B=${$y_j|1\leq j\leq 6$} we consider the map $f:A\to B$ defined by:
$$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....
0
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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
votes
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 ...
-1
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 \...
