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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5answers
74 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
44 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
282 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 ...
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1answer
28 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
10 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
11 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
25 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
35 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
17 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
37 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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0answers
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
31 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
76 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 ...
1
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1answer
24 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
12 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
20 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 ...
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1answer
13 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 ...
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1answer
21 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 $xf = x + 1,xg=x^2$, ...
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0answers
11 views

Compute the composition of functions

I got wrong in this very odd question for my assignment. Can somebody help me with the answer of this question provided an explantion? Thank you a lot in advance!
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1answer
28 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
20 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 ...
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1answer
12 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 ...
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1answer
54 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
42 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 ...
4
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1answer
39 views

Prove that two groups of functions are isomorphic

The two functions $f(x)=\frac{1}{x}$ and $g(x)=\frac{x-1}{x}$ generate, with the operation of function composition, a group $G$ of functions. Prove that this group is isomorphic to the group $S_3$. I ...
8
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4answers
1k views

If the absolute value of a function is continuous, is the function continuous?

If $|f(x)|$ is continuous at $a$, is $f(x)$ continuous at $a$? I tried doing it using composite functions. If $g(x)= |x|$, then $g\circ f(x)= |f(x)|$. Since $g(x)$ and $g\circ f(x)$ are continuous, ...
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1answer
38 views

General method for composition of piecewise defined functions

There is a similarity in questions about composition of functions piecewise defined (see e.g. here, here and here). In these questions the goal is always the same: Given $f,g$ piecewise defined, ...
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1answer
15 views

How to alter the interval of a composite function

Let $f, g : R → R$ $$f(x) = \begin{cases} x + 3 &\text{if } x ≥ 0,\\ x^2 &\text{if } x < 0 \end{cases}$$ $$g(x) = \begin{cases} 2x + 1 &\text{if } x ≥ 3,\\ x ...
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2answers
17 views

Manipulation of Composition of Functions

Let $f: A\to B$ and $g,g' : B \to C$. Prove that if $g \circ f=g' \circ f$ and $f$ is surjective then $g=g'$. Is it fair to say that since $f$ is surjective it has a right inverse such that $f\circ ...
0
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1answer
61 views

Composite function of $R$-module

Let $M$, $N$, $P$ be $R$-module and functions $f:M\rightarrow N$, $g:M\rightarrow N$, $h:N\rightarrow P$, $k:P\rightarrow M$, which need not be homomorphisms. Define $f+g:M\rightarrow N$ by ...
1
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2answers
41 views

What is the 'formula' of a composite function?

Consider $f: \mathbb{R} \to \mathbb{R}$ such that $f(x) = \frac{1}{x^2 +1}$ and $g: \mathbb{R} \to \mathbb{R}\times \mathbb{R}$ given by $g(x) = (3x, x^2)$. I was asked to find the 'formulas' of $f ...
2
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1answer
78 views

What's the shape of this “addsTo” function …?

Note that in this combinatronics question, How many lists of 100 numbers (1 to 10 only) add to 700? I was asking: For an array of 100 numbers, each 1 to 10 inclusive, and the total is T - how many ...
1
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1answer
30 views

Notation for repeated composition of functions

I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ \cdots \circ {\tau _n}(z)$ By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n ...
0
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1answer
14 views

How do I find and list compositions for (f) and (g)?

Ok, I've literally just spent the last 2 hours just to figure out two compositions problems for homework, and I've about had it. Anyone here that can help? Problem 1 $$ f(x) = 2x(2) - x -3 $$ $$ ...
2
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1answer
47 views

Function composition proof

One of the questions assigned to me for homework was: Is it true that $ f\circ (g \circ h ) =f \circ g + f \circ h$? I am in the understanding that $f \circ g$ means $f(g(x))$, so $ f\circ (g ...
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2answers
48 views

The limit of composition of two functions

I need your help in solving this limits problem. Let $f$ and $g$ be two functions defined everywhere. If $\lim_{u\to b} f(u) = c$ and $\lim_{x\to a} g(x) = b$, then you may believe that ...
0
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0answers
25 views

Composition operator of a non-analytic function

I recently came across a problem that can be reformulated in simple terms involving the composition operator $$ C_g\colon X \to X, \quad f \mapsto f \circ g $$ for functions $$ f,g\colon \mathbb{C} ...
4
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1answer
60 views

Simplify an iterated function

If we iterate the function $f(x) = \ln(x + 1)$, we get: $$f(f(x)) = f^2(x) = \ln(\ln(x + 1) + 1)$$ $$f(f(f(x))) = f^3(x) = \ln(\ln(\ln(x + 1) + 1) + 1)$$ $$f(f(f(f(x)))) = f^4(x) = \ln(\ln(\ln(\ln(x + ...
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2answers
23 views

Composition of Functions from $\mathbb{R}$ to $\mathbb{R}^2$

Is the composition of functions from $\mathbb{R}$ to $\mathbb{R}^2$ a well defined notion? I was asked whether or not the composition of such functions constitutes a binary operation, but I don't know ...
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1answer
47 views

On the existence/applications of infinitely-nested functions

Inside a previous question, one particular nested function shown is the known tetration. This "kind" of arbitrary repeated functions has always intrigued me, because inside their properties lie so ...
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1answer
34 views

Composition of Ordered Pair

I'm doing math exercises from a Computer Science book and I am confused as to how the following result (from the solutions manual) is obtained: Given the function f={(a,b), (a,c), (c,d), (a,a), ...
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0answers
36 views

Decomposition of integer polynomials $P(x)$

Let $P(x)$ be an integer polynomial of composite degree $D$. A decomposition is the opposite of a composition. For instance composition of polynomials $A(x),B(x)$ gives $A(B(x)) = C(x)$. The ...
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2answers
58 views

What does let $F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$ mean?

Let $F$ denote the set of all functions from $\{1, 2, 3\}$ to $\{1, 2, 3\}$. I'm supposed to prove that this statement is true or false, $$∀f ∈ F, \;∃g ∈ F\tag i$$ so that $g(f(1)) = 2$ But I'm not ...
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0answers
64 views

Calculus - Proof composition of two function.

$f,g$ are defined in $\mathbb R$ and these conditions apply. $$1) \lim_{x \to x_0} g(x) = y_0$$ $$2) \lim_{y \to y_0} f(y) = L (L\in\mathbb R)$$ $$3)\mbox{ There is a pocked enviroment of }x_0 \mbox{ ...
2
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2answers
83 views

Proof for Surjections

I'm reading through Basic Algebra I (which I enjoy so far. Thoughts on this for self-studying?) and am having a difficult time proving surjection. I believe I understand the concept, but when it comes ...
3
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2answers
61 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
2
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0answers
130 views

Rendering the derivative of composite functions from a graph

I'm on a workbook problem and I want to make sure I'm doing it properly. The problem asks me to find the derivatives of composite functions when given only the graphs of the original functions, here ...
3
votes
2answers
35 views

Composition of convex and power function

Let $g$ be a convex nonegative function, and $p\ge1$. To show: $f(x)=g(x)^p$ is convex. Let $h(x)=x^p$. Then clearly $f=h \circ g$. Denote $\tilde{h}$ as the extended-value extension of $h$, which ...
0
votes
2answers
57 views

If $g \circ f$ is injective, so is $g$

If $g \circ f$ is injective, so is $g$ I don't think this is true. I think that $f$ has to be surjective. So I am going to try to prove that: If $g \circ f$ is injective, and $f$ is ...
0
votes
1answer
38 views

Composition of functions.

If $h(x) = 2x + 1$, find $h(x + 2)$ ...and a second question. Given $f(x)=x^2+1, g(x)=x-2$, then what is the domain of $f(g(x))$ $\begin{align}\text{Work: } f(g(x)) & =(x-2)^2+1 \\ & ...