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Questions tagged [function-and-relation-composition]

For questions about the composition of functions and relations.

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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
36 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 ...
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2answers
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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 ...
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3answers
212 views

If composition of two functions $f,g$ is continuous, does that imply continuity of both $f$ and $g$? [closed]

If the composition $f\circ g$ of $f$ and $g$ is well defined and is continuous, does that necessarily imply that $f$ is continuous and $g$ is continuous?
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1answer
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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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24 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 ...
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1answer
40 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 ...
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2answers
23 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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1answer
71 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{-...
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1answer
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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 ...
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1answer
92 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 ...
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2answers
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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
58 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 ...
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2answers
120 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
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1answer
64 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
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1answer
16 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 $R$ ...
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1answer
21 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 \...
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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 ...
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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
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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
25 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
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1answer
30 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 \...
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0answers
21 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 ...
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0answers
28 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 ...
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1answer
41 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 ...
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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] \...
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2answers
71 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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1answer
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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 ...
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3answers
44 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 ...
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1answer
29 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: ...
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1answer
63 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 ...
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4answers
49 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 ...
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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 \...
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3answers
14 views

Find the composition f(g(x)) when four functions are given

Determine $f \circ g$ for the following functions: $$ f(x) = \begin{cases} -x, & x < 0\\ x+1, & x \ge 0 \end{cases} \quad \text{and} \quad g(x) = \begin{cases} x^2, & x \le 2\\ x+2, &...
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2answers
21 views

How to find domain of complicated composite functions

If I was asked to find the domain of arccos($e^x$), are there universal steps I can take to be able to find the domain? I know that you want the inner function in f(g(x)) to be defined as well as the ...
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0answers
13 views

Forming a relational composition and justifying its properties

So I have two relationships $Visited \subseteq Student$ X $Country$ $ParentsFrom \subseteq Student$ X $Country$ Visited = { (Mark, Poland), (Mark, UK), (Rick, USA), (Rick, Canada), (Rick, Uganda), (...
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1answer
340 views

Number of real zeroes of iterated polynomial: $x^3-2x+1$

If $P(x)=x^3-2x+1$, define $z_n$ as the number of real roots of the polynomial $P^{\circ n}(x)$, where the superscript denotes $n$-fold composition. Can we find a general formula for $z_n$, or perhaps ...
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2answers
36 views

Composite function derivatives

Is this statment right in derivative of compoaite three functions $(f \circ g \circ h)'(x) = (f\circ g)' (h(x))$
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3answers
21 views

Prohibited values when plugging one formula into another?

Say f(x)=$\frac{1}{x}$ and g(x)=$x-5$ and we where to plug g into f as a composite function, would x = 0 still remain a prohibit value of f or not? Thanks!
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1answer
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Number of real roots of an iterated quadratic: $x^2-3/2$

I was messing around with polynomials and their real roots when I, as recreational mathematicians do, asked myself the following random question: Suppose I am given a polynomial $P(x)$. How can I ...
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4answers
62 views

All values of $m,b \in \mathbb{R}$ such that $f(x) = mx+b$ is a solution to $f(f(x)) = x$

I am asked to find all values of constants $m,b \in \mathbb{R}$ such that $f:\mathbb{R}\to \mathbb{R}, x\mapsto mx+b$ is a solution to $f(f(x)) = x$. So since $f(f(x)) = m(mx+b) + b = m^2x + mb + b$, ...
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1answer
24 views

Composition of functions of given set

If ∀x ∈ {1,2,3,4} , $f$(x) = $ x^2 $ and ∀x ∈ {2,4,3,6} , $g$(x) = $ x+1 $. Find $ (g\; o\; f ), (f\; o\; g)\; and \;Im(f \;o \;g)$ I dont understand the question much. If someone can help me it ...
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1answer
58 views

What is a function $g$ called, which, composed with a certain function $f_2$, yields a given function $f_1$ (i.e. $f_1 = f_2\circ g$)?

Let $X_1$, $X_2$, and $Y$ be non-empty sets. Let $f_1:X_1\rightarrow Y$ and let $f_2:X_2\rightarrow Y$. What's the terminology for a function $g:X_1\rightarrow X_2$ that satisfies: $f_1 = f_2\circ g$? ...
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1answer
84 views

Proving f is onto given if $g \circ f = h \circ f$, then g = h

I realize that there has already been an answer to this problem. But I want to know if my proof was correct. Thank you for your time. Problem Suppose A,B, and C are sets and $f: A \rightarrow B$ ...
2
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2answers
47 views

Is there any unique function $c=f(a, b)$ such that if I know only $c$, I can deduce unique $a$ and $b$ from it? [duplicate]

I want to know a relation $c = f(a, b)$ (where $a, b, c$ are real numbers) such that if I only know $c$ I can deduce $a, b$ inputs from it. As an example consider $f(a, b) = a+b$, then $f(2,3) = 5$ ...
1
vote
1answer
42 views

Decomposing a binary function, into (1) a continuous function and (2) a thresholding function.

Suppose $f$ is an arbitrary binary function and it is given to us: $$ f: \mathbb{R}^{d} \rightarrow \{0, 1\} $$ Prove that it is always possible to write this function, as a composition of two ...
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0answers
32 views

Seeing the tendency of functions

If $f:[1, 10]\to [1,10]$ is a non decreasing function and $g:[1, 10]\to [1,10]$ is a non increasing function. Let $h(x) =f(g(x)) $ with $h(1)=1$. Then $h(2)$: $\text{(A)}$ lies in $(1,2)$. $\text{(B)...
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1answer
34 views

Function composition - solving for functions

I have a simple, continuous, real-valued function $\sigma$, whose functional form I know, and I know that two other invertible functions, $f$ and $g$, satisfy the following relationships: $f\circ\...
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1answer
38 views

What are examples of pseudorandom functions that can be preformed without the assistance of a computer?

I'm trying to generate a seemingly random list of integers without the use of a computational device; one capable of being unraveled with a mathematical function. What's the best way to devise a ...