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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19 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.
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1answer
128 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 ...
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4answers
265 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 ...
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3answers
40 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?
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3answers
36 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 ...
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0answers
13 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 ...
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1answer
22 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. ...
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0answers
26 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 $f(x)=\begin{cases}\frac x 2 &, x\in\mathbb N_{even}\\ x+9 &,x\in\mathbb N_{odd}\end{cases}$ ...
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1answer
31 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 ...
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1answer
15 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 ...
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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 ...
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1answer
23 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?
3
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1answer
31 views

Is there a multiple function composition operator?

Is there a commonly-accepted operator which defines multiple function composition? I have not been able to find one on any of the related Wikipedia pages. In one of my proofs, I've been finding it ...
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0answers
21 views

Regarding functions composition in descrete math

Let $f : A \to B$ be a function. Let $g:B \to B$ and $ℎ:B \to B$ be functions that are total surjective and injective (bijection). Prove that if $h \circ g \circ f$ is total and injective, then $f$ is ...
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2answers
18 views

Domain and Range of Function Composition

Given: $a\left(x\right)=e^x$ $b\left(x\right)=\left|x+2\right|$ $c\left(x\right)=\frac{\left(x-2\right)}{\left(x+1\right)}$ What is: $\left(\frac{a\cdot ...
0
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1answer
36 views

How to check convexity of a composition when some properties of inner and outer functions are known?

If $g(x)$ function is concave in $x$, and we want $g( f(x) )$ (where $f(x)$ is another function) to be convex in $x$, what are the required properties of $g(x)$ and $f(x)$? It would be appreciated ...
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1answer
28 views

Composition of Invertible Functions

Once again we're studying domain and range in class and I encountered this problem. If $f(x)$ and $g(x)$ are both invertible functions, and the domain and range of each function is the set of real ...
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2answers
35 views

Linear composition

can you help me with this quest? About composition $f$ and vector space $\mathbf{V}=\mathbb{Z^4_2}$ we know the following: $f \circ f = id_V$,$~~f $ $ \left(\begin{array}{ccc} 1\\ 0\\ 1\\ 0\\ ...
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3answers
55 views

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true?

let $f$ and $g$ be two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing. Which of the follwing is true? (a). If $g$ is continuous, then $f\circ g$ is continuous. ...
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0answers
44 views

Is this proof show that the derivative of zeta function has no zeros in the critical strip?

suppose that :( $k \circ \zeta )(s)$= $(\zeta \circ k )(s) \neq 0 $.....$(1)$ ,and suppose that $ k(s)=\zeta(1-s)$ where : $\xi(s) = s(s-1) \pi^{s/2} \Gamma\left(\frac{s}{2}\right) \zeta(s)$. and ...
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1answer
30 views

Is the composition of a set of convex functions convex?

Here we see the proof for $f(x)$ being convex where $$f(x) = h(g(x))$$given $h$ is convex and nondecreasing and $g$ is convex. But what if $$f(x) = h(g_1(x),g_2(x),g_3(x),...,g_k(x))$$ where each ...
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0answers
27 views

Fourier Transform of a function with sinusoidal sampling

What is the relation between the Fourier Transform (FT) of $f(x)$ with regular sampling and the FT of $f(x)$ with sinusoidal sampling? In other words, it's a FT of a function composition $f\circ ...
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0answers
21 views

Function composition check and define the resulting function

Here is the question I need to do: Consider the following functions: $$\begin{align}f:\Bbb N\to\Bbb B&\text{ defined by }f(x)=x>8;\\g:\Bbb N\to\Bbb N&\text{ defined by }g(x)=(x\cdot 3) ...
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5answers
151 views

Composition of Inverse Functions

$f$ and $g$ are inverses of each other when $f(g(x)) = x = g(f(x))$. However, can there be 2 functions where $f(g(x)) = x$ but $g(f(x))$ does not equal to $x$? I feel like there are but I cannot find ...
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0answers
37 views

Does each element of $D4$ have an inverse in $D4$?

We are just starting the concept of permutations of objects in my class and I'm having trouble to grasp this particular question. I'm assuming it does have an inverse because of all the different ...
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3answers
48 views

If $f: A→B$ and $g: B→C$ are surjective, then $g\circ f$ is surjective.

In my homework, I wrote: Assume f and g are surjective. Let m be an element of C. then there exists a b that's an element of B, such that g(b) = m and an a element of A such that f(a) = b by ...
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1answer
12 views

Jacobian of composite functions with different number of variables

It is said that it is possible to calculate the Jacobian of a composed function by multiplying the Jacobians of each function, that is $$ J_f = J_{f_1} \cdot J_{f_2} \cdots J_{f_nx} $$ where $$ f = ...
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5answers
81 views

What is an example of a bijective function f: Z to N that isn't piecewise?

Like without using if even or odd. Like how you can define a bijection $f\colon\mathbb{N}\to\mathbb{Z}$ by is $f(n)=\lfloor n/2\rfloor\cdot(-1)^n$.
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1answer
49 views

Prove that the set of all functions is not a group under function composition.

Consider the set $F$ of all functions from $\{1,2,3\}$ to $\{1,2,3\}$. There are $3^3= 27$ of them. Prove this set is not a group under function composition. I thought that it violates the ...
3
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5answers
364 views

Chain rule with triple composition

We are supposed to apply the chain rule on the following function $f$: $$ f(x) = \sqrt{x+\sqrt{2x+\sqrt{3x}}} $$ I assumed we could rewrite this as $$ f(x) = g(h(j(x))) $$ However, I was not sure ...
0
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1answer
53 views

composition of uniformly convergence sequence with continuous function, is uniformly convergence?

Let $(f_n)$ be a series of functions in $C[0,1]$ that uniformly converge to a continuous function $f\in C[0,1]$. a. Let $g: [0,1]\to [0,1]$ be a continuous function. Is it true that $f_n\circ g$ ...
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0answers
13 views

Asymptotics and function composition

In the following question: Big O and function composition It is explained that if $a, b, c, d$ are functions and $a = O(c), b = O(d)$ it doesn't mean that $a ∘ b = O(c∘d)$. However, what if we allow ...
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0answers
14 views

Composition of multivariate real rational functions

Consider an $n_1-$variate real reduced rational function$f_1$ of degree $d_1$ and an $n_2-$variate real reduced rational function $f_2$ of degree $d_2$(sum of degrees of numerator and denominator). ...
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2answers
32 views

Transpose of composite functions

I want to prove that $(gof)^T$=$f^Tog^T$ where $f,g$ are linear maps. I know that I can just use definitions but I don't know exactly how. Can anyone point me in the right way?
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4answers
40 views

If an iterated function $f \circ f$ is the identity function, is $f$ an identity function also?

If we have $f: \{1, 2, 3\} \to \{1, 2, 3\}$ and $f \circ f = id_{\{1,2,3\}}$ is the following then always true for every function? $f = id_{\{1,2,3\}}$
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0answers
19 views

How to compose these maps?

I am having a composition of two maps: $$ f:\mathbb{R}\rightarrow\mathbb{R_0^+},f(x)=x^2 $$ $$ g:\mathbb{R_0^+}\rightarrow\mathbb{\mathbb{N_0}},g(x)=\lfloor x\rfloor $$ $$h=g\circ ...
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1answer
45 views

Finding all continuous and discontinuous points of composite functions

Let $f(x) = \operatorname{sgn}(x)$ and $g(x) = 1 + x^2$. How do I go about finding all the continuous and discontinuous points of the functions $f\circ g$ and $g\circ f$ ?
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12 views

Range of a function composition is a subset of the range [duplicate]

Let $L:\Bbb R^n → \Bbb R^m$ and $M:\Bbb R^m → \Bbb R^p$ be linear mappings. Prove that $Range (M◦L)$ is a subspace of $Range (M)$. So I began by defining: $Range (M◦L)$ is a subset of $\Bbb R^p$ ...
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1answer
34 views

Commutative Diagrams and Polynomials

Recently, considering how algebraic numbers may be defined by the algebraic properties that they satisfy (and in particular, the polynomials of which they are roots), I started to wonder about, for ...
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1answer
110 views

Range$(M◦L)$ is a subspace of Range$(M)$

Define the following linear mappings: $$L:R^n→R^m$$ $$M:R^m → R^P$$ Prove that Range $(M◦L)$ is a subspace of Range $(M)$. What I have so far (not sure if correct): Range $(M◦L)=R^p$ and Range ...
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1answer
27 views

Composite Functions

$f(x)= \dfrac{1}{10x+17}+13$ $g(x)= \dfrac{1}{9x-6}$ I need to find $f(g(x)).$ How do I do this? I keep on getting it wrong. The correct answer is $\dfrac{1998x-1202}{153x-92}$. But I am unsure how ...
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1answer
14 views

Finding inverses of two functions and their compositions to solve for unknown.

$$f(x) = 23x + 27,\;\; g(x) = 12x - d$$ I've found $f^{-1}(x),$ and $\,g^{-1}(x)$, but I don't know how to solve for $d$, given $$f^{-1}(g^{-1}(x)) = g^{-1}(f^{-1}(x)).$$ How do I do this please?
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2answers
23 views

Which of the following is constant?

If $f,g$ are continuous real valued functions such that $f\circ g$ is constant then which of the following must be constant? $$f,g,g\circ f$$ I think when $f\circ g$ is constant then at least one of ...
0
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1answer
18 views

Composing Piecewise Functions

I was wondering how to compose piecewise functions. On a practice exam I was reading, a question asks what F(F(x)) will look like if F(x)= 2x if x<1/2 and = 2-2x if x>=1/2. Would I just ...
0
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1answer
22 views

Mapping of elements notation - Cohn - Classic Algebra Page 13

So Cohn uses the notation that many have wanted to change to, being $xfg$ rather than $g(f(x))$, and I have had the example: Let $f,g: \mathbb{N} \to \mathbb{N}$, be given by $xf = x + 1,xg=x^2$, ...
0
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1answer
30 views

Limit involving the square root and composition

I want to show that if $\lim_{x \to x_0} f(x) = L > 0$, then $\lim_{x \to x_0} \sqrt{f(x)} =\sqrt{L}$. I'm at the point where I have $|\sqrt{x} - \sqrt{x_0}| < \delta$, $\forall \delta < ...
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0answers
24 views

Prove that the composition of two “closed form functions” is itself a “closed form function”?

So I have been given the definition of a "closed form function" that is a set of functions built inductively (mapping from and to the complex) starting with the fact that the constant functions ...
0
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1answer
13 views

Simple composition of linear maps

Let $F : R^2 → R^2$ s.t. $x → (−x_2, x_1)$ and $G: R^2 → R^2$ s.t $x → (x_2, sin x_1)$. Evaluate $G ◦ F$ and $F ◦ G$. I have said $(G ◦ F)(x) = G(F(x))=-x_2,sinx_1$, but I feel as though this is ...
0
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1answer
67 views

Finding limits of composition functions of a piecewise?

Having insane amounts of trouble doing this. Heres a graph of $f(x)$: How am I to calculate $\lim_{x \to 0} f(f(x))$ , $\lim_{x \to 3} f(f(x))$ , $\lim_{x \to 0} f(1+x^2)$. One that is even more ...
0
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1answer
47 views

How to calculate the limit of f(f(x)) if f(x) is a piecewise function?

Say f(x) is some piece wise function, and on some interval it is a quadratic equation. How do we determine $$\lim_{x \to -2} f(f(x))$$ Would it just be first we sub the quadratic inside the quadratic ...