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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1answer
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Compostion of functions in taylor series

I was attempting to help answer a question, but I am curious about the following. Suppose we have an analytic function $f(z)\equiv\sum\limits_{k=0}^{\infty} a_k(z-z_0)^k $ on some suitable disk of ...
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0answers
48 views

Don't know when to add negative numbers

I'm definitely not a math person and only did general mathematics in high school, and unfortunately, not paying as much attention to that as I should have. Well, I'm doing Discrete Mathematics in my ...
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1answer
52 views

How to solve for $f(2)$ given $3f(x)+f(2-x) = 2x^2$?

I just came across the problem where I'm given that $3f(x)+f(2-x) = 2x^2$ and I need to find $f(2)$. I know it is simple, please help me.
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1answer
70 views

Surjectivity of Composite Functions

The question I'm asking might be rather simple, but I couldn't find relevant information (maybe it's too trivial?). Here's the question that baffled me. Let $f:X\rightarrow Y$ and $g:Y\rightarrow ...
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1answer
31 views

Composition with exponent Sobolev Space

I have a problem with following statement: We have $f \in W^{1,1}(B) (B-ball\ in\ \mathbb{R^n}), \ \nabla f \ e^{f} - $ integrable $\Rightarrow \ e^f \in W^{1,1}$ I've started with a sequence of ...
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5answers
172 views

Application of Composition of Functions: Real world examples?

Do you know of a real world example where you'd combine two functions into a composite function? I see this topic in Algebra 2 textbooks, but rarely see actual applications of it. It's usually plug ...
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1answer
82 views

Prove the continuity in the composition function.

If $g$ is continuous at $c$ and $f$ is continuous at $g(c)$ Then prove that $(f\circ g)$ is continuous at c. To prove this I have done something: Given: $$\lim_{x\to c}g(x)=g(c) \tag 1 $$ ...
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3answers
27 views

Notation for function compositions/derivatives

When given $(f \circ g)'(0)$, does it mean to compose the 2 functions first, then take the derivative of the composed functions and evaluate it at $0$, or take the derivative of $g$ first and evaluate ...
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39 views

How to compose vector-valued functions

Let $f(u,v) = (uv, u+v)$ and $g(x,y) = (e^{xy}, x-y)$. Calculate $f \circ g$. I don't understand how to compose these functions together. The question does not make any sense to me at all whatsoever, ...
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2answers
69 views

Please solve the equality of this function.

Let $f,g,h:\mathbb R\to \mathbb R$. Show that: $$ (f+g)\circ h = f\circ h + g \circ h $$ $$ (f\cdot g)\circ h = (f\circ h)\cdot(g \circ h) $$ I know that $(f+g)(x)=f(x)+g(x)$. But I don't know how ...
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2answers
28 views

$n$ composition of functions when $n \to \infty$

Let $x \in \mathbb{R}$ and $n \in \mathbb{N}$. Let $f(x)$ be continous over the whole domain of $a<x<b$. Let the composition of functions $f^{(n)}(x) =f(f(...f(x)))$. Let $g(x)$ defined by ...
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0answers
42 views

General (Set Builder) definition for a relation composed with itself n times

Questions What does the set builder notation for $S\circ R$ look like? I'm having the most trouble knowing when there is too much information or not enough information on either side of the 'such ...
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0answers
34 views

Differentiablilty of composition functions

Two questions I suppose. One comes from a test I recently took that I didn't quite get/understand the method I should be using (or even how I should proceed) Let $f:R^2 \rightarrow R$ s.t $f$ is an ...
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1answer
43 views

I can't understand this question.

the function f is defined by f(x)=m+x/2+3x for all value of x except when x=h.Find the value of h .
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2answers
32 views

Fractional Composite of Functions

I would like to know how I can calculate a fractional composition of a function. Let be $f(x)$, where $x \in R$ and $f(x) \in R$. I now how to do $f(f(x))=f^2(x)$. Now suppose I would like to do ...
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1answer
29 views

Function composition in computability

I have been reading Cutland's computability book, which is really good! However, I have found myself thinking way too much about one little passage in the the third section of the second chapter (the ...
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4answers
81 views

Let $f: A\rightarrow B$ and $g: B\rightarrow C$ be invertible maps, show that $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

I am working on the following problem for my abstract algebra class, and I wanted to get some feed back to see if I am on the right track. Here is what I have so far. Let $f: A\rightarrow B$ and $g: ...
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1answer
46 views

Write the function as a combination of elementary functions

$2xe ^{(-4x^2)}$ Is this correct? $f(x) = -4x^2, g(x) = e^x, h(x) = 2x$ $h(x)\cdot g(f(x))$
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1answer
30 views

Composition of a piecewise and non-piecewise function

Say you have 2 functions, one of which being a piecewise function: $f(x)= x^2+2, x<1$ or $2x^2+2, x>=1$ And the other: $g(x)=x^4+1$ How would you find the $f[g(x))]$? I understand regular ...
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1answer
73 views

Continuity of $h(x)=f(x) \cdot g(x)$

$h(x)=f(x) \cdot g(x)$ I want to check whether this function is continuous in its domain $\mathbb{R}$ or not. definition by cases: $f(x)$ and $g(x)$ are both continuous $\Rightarrow f(x) \cdot ...
2
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1answer
43 views

Does(n't) associativity of functional composition follow straightaway from associativity of relational composition?

One thing I find puzzling about the typical way in which associativity of functional composition is proved is that it makes explicit use of the fact that a function is a 'right-unique' relation, i.e. ...
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2answers
73 views

Maximal domain for composite functions.

Question $\mathbf 5$ If $f:(-\infty,1)\to R$, $f(x)=2\log_{\,e}(1-x)$ and $g:[-1,\infty)\to R$, $g(x)=3\sqrt{x+1}$, then the maximal domain of the function $f+g$ is ...
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2answers
69 views

How to find $f(x)$ and $g(x)$ when only given $f(g(x))$

I've learned how to find $f(g(x))$ when given the two $f(x)$ and $g(x)$ functions fairly easily, but I haven't found anywhere online showing how to do the opposite. For this question I'm working on ...
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3answers
62 views

Solve the composition $f \circ g=?$ and $g\circ f=?$

a) $f(x) = \sqrt[3]{x}\ $ and $g(x) = x^3$ find $f\circ g=?$ and $g\circ f=?$ I have $f\circ g = f(g(x)) = f(x^3) = \sqrt[3]{x^3} = x$ So basically, first we replace $g(x)$ with its value, then ...
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3answers
33 views

Proofs regarding composition of functions

I'm having trouble approaching the following question: Is the following statement true or false, provide a proof or a counterexample. If $h: A\rightarrow B, \ g: B\rightarrow C, \ f: B\rightarrow ...
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2answers
74 views

function composition - n times

Please consider this function: $$f(x) = \frac{x}{{\sqrt[6]{{1 + {x^6}}}}} $$ What would be the value of the composition (n times): $$f \circ f... \circ f = ? $$ I tried doing it manually, maybe ...
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1answer
33 views

Function composition and inverse

Consider f : ℝ \ {1} → ℝ \ {1} given by f(x) = x/(x-1) I need to find: 1) f ◦ f ◦ f and 2) the inverse function f^-1(x) So far I have: 1) f(f(x/(x-1)) = f(x) = x/(x-1) which is suspicious to me ...
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2answers
490 views

Find Domain and Range of Composite Function

Given $$f(x) = \{(-5,0),(-4,-2),(-2,3),(1,5),(4,2)\}$$ $$g(x) = \{(0,-2),(8,4),(-2,5),(5,-5),(3,1)\}$$ $1$) Find the domain and range of $f(g(x))$. $2$) Find the domain and range of $g(f(x)$.
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What can I use as the generic term for “a function that is composed with another”?

Suppose I am talking about the composition $g \circ f$ (or more generally $f_n \circ \cdots \circ f_1$). Is there a generic term for the functions $f$ and $g$ (the functions $f_i$)? "Compositand"?
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1answer
56 views

If two compositions are bijective, then all functions involved are bijective?

Given functions $f:A\to B$, $g:B\to C,$ and $h:C\to D.$ Provided $g\circ f$ and $h\circ g$ are bijective, prove each of the functions $f$, $g$, and $h$ is bijective.
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1answer
24 views

Proving composition of functions

I am trying to prove the following theorems: Let A, B, and C be nonempty sets and let $f : A \rightarrow B$ and $g : B \rightarrow C$. If $g \circ f : A \rightarrow C$ is an injection, ...
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0answers
51 views

Example of sets $A$ and $B$ and functions $F$ and $G$ such that $F: A \rightarrow B, G: B \rightarrow A, G \circ F = I_{A}$, and $G \neq F^{-1}$

Give an example of sets $A$ and $B$ and functions $F$ and $G$ such that $F: A \rightarrow B, G: B \rightarrow A, G \circ F = I_{A}$, and $G \neq F^{-1}$ I was thinking maybe $F$ can be a function ...
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2answers
879 views

composition of two uniformly continuous functions.

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ and $g : \mathbb{R} \rightarrow \mathbb{R}$ are two uniform continuous functions. Which of the following options are correct and why? $f(g(x))$ is ...
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1answer
25 views

Simplify functions involving modular arithmetic

In this question, the answer says that f o g(x) = x. But I am unable to get this result. The expression I am able to get is that f o g(x) = 7*(x mod 3) + 57*(x mod 7) (mod 21). I am unable to ...
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2answers
141 views

$A={1,2,3,4,5}$. How many functions $f : A \rightarrow A$ so that $f\circ f (1) = 3$

A={1,2,3,4,5} How many functions f : A -> A so that f is onto? 5! is this correct? How many functions f : A -> A so that $f\circ f$ (1) = 3? f(1)=1 f(1)=2 f(1)=3 f(1)=4 ...
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3answers
170 views

Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?

True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
0
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1answer
118 views

MathCAD: composite function, (fog)(x)

So sorry for the dumb question, but I can't figure this out and I have been searching for the answer and can't find it... and at this point I no longer know what to Google. Using MathCAD, how do I ...
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2answers
73 views

Composition of Identical Functions

I have come across a problem which asks to find $f(x)$ such that $f(f(x))=-x$. Nothing I can find has anything pertaining to the composition of two identical functions. Is there a way that I can ...
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1answer
53 views

How do I prove the definition of a homomorphism?

The question is asking me to prove that $f(a \circ b) = f(a) \circ f(b)$. This I believe is referencing our previous proof which tells us: Assume $g:x \rightarrow y $ is a bijection and for an $a ...
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0answers
55 views

Repeated function composition

I have a recursive function repeat, that composes n calls to f with a start value ...
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1answer
44 views

How do i find the values of these functions?

Suppose $f$ and $g$ are one-to-one functions such that $f(2)=7$, $f(4)=2$, and $g(2)=5$. If possible, find the values of A) $(g \circ f^{-1})(7)$ B) $(f \circ g^{-1})(5)$ C) $(f^{-1} \circ ...
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2answers
47 views

Is this a valid example for $f \circ g = id_M$, but $g \circ f \ne id_N$

The exercise is to give an example for two sets $M$ and $N$, and functions $f$ and $g$, for which $f \circ g = id_M$, but $g \circ f \ne id_N$. My idea is a bit based on my computer programming ...
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0answers
23 views

Restricting binary relations by composing with an “inclusion binary relation”

If $X' \subseteq X$ then we may define an inclusion map $\iota : X' \to X$ where $\iota(x) = x$. One use of $\iota$ is that we can express the restriction of some $f : X \to Y$ to $X'$ as $f|_{X'} = f ...
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1answer
51 views

Introduction to Analysis: Properities of Functions

If I remember correction from my abstract algebra course, if $f(x)$ is defined for all x and is bounded, then composition mapping $f\cdot g$ is also bounded, and so should $g\cdot f$ since the range ...
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1answer
75 views

Determining if a homomorphism is an isomorphism

Let $T \in \mathcal{L}(V)$, where $\mathcal{L}(V)$ is the set of linear operators mapping a vector space $V$ to itself, and let $U$ be an isomorphism from $V$ to another vector space $W$. We claim ...
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2answers
114 views

Real Analysis: Continuity of a Composition Function

Suppose $f$ and $g$ are functions such that $g$ is continuous at $a$, and $f$ is continuous at $g(a)$. Show the composition $f(g(x))$ is continuous at $a$. My idea: Can I go straight from definition ...
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1answer
141 views

Composition of Periodic Functions.

Suppose $f(x)$ is defined for all $x$; then $f \circ \cos x$ is periodic and $2 \pi$ is a period. Conversely, if $g(x)$ is defined for all $x$ and is periodic, with $2 \pi$ as a period, can one find a ...
0
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1answer
29 views

Calculating Indefinite integral

Suppose $f(x)=\int g(x)dx+C_1$, Then how can I get $f(a-x)$? I thought $f(a-x)=\int g(a-x)d(a-x)+C_1=-\int g(a-x)dx+C_1$. But it seems not correct. Thanks.
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3answers
60 views

Composite bounded functions

Prove $f(x)$ is bounded $\rightarrow$ that $f(g(x))$ is bounded. For all x in $f(x)$ ang $g(x)$. To my understanding, suppose $f(x)$ is bounded, then do we need to show that the composition function ...
0
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1answer
42 views

How do I solve this function and find its domain?

Suppose that $f(x)= -x^2+1$ and $g(x)= \sqrt{x}$. How do we find $f \circ g$ and $g \circ f$ and their domains?