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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2
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2answers
74 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
42 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
22 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
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2answers
23 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
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2answers
48 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
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1answer
35 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 \\ & ...
2
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1answer
20 views

Domain of a composite function

I was given the question: Find the domain of the function $f(x)=\ln(\ln(\ln x))$ I found the answer by inspection: $\qquad D(\ln x)=(0,\infty)$ $\therefore\quad D(\ln(\ln x))=(1,\infty)$ ...
0
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2answers
38 views

Prove Lipschitz function composed with an integrable function is integrable on [a, b]

Given a Lipschitz function $g$ (i.e. $|g(x) - g(y)| \leq L |x - y|, \forall x, y \in dom(g)$), and an function $f$ integrable on $[a, b]$, how do we prove $g \circ f$ is integrable on $[a, b]$, ...
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0answers
10 views

Proving IVP with composition equality.

Let $a<b$ for $a,b$ in $\mathbb R$. Let $f,g:[a,b]\to[a,b]$ be two continuous functions such that $f\circ g=g\circ f$. Show that there exists an $x$ in the interval such that $f(x)=g(x).$ I've ...
0
votes
1answer
36 views

Function Decomposition

How do I decompose a function when I'm given $f(g(x))$ and $f(x)$ and the required is $g(x)$? I done some searching on Google and most sites demonstrate the solution where it's left open, they just ...
0
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2answers
24 views

Taking the compositions of two constant functions

The questions asks to prove that the composition of g with f is not equal to f with g. However, I don't know whether you can even take the composition of constant functions or how. so if f(x)=2 and ...
1
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2answers
54 views

Composition of non-monotonic convex function

Given the following composition of functions: $h:\Bbb R^k\rightarrow\Bbb R$ $g:\Bbb R^n\rightarrow\Bbb R$ $f(x)=h(g_1(x),g_2(x),...,g_k(x))$ There are known rules which guarantee ...
0
votes
1answer
26 views

Need help with compositions of relations

Prove that given relations $R_1 \subseteq A \times B$, $R_2 \subseteq B \times C$, $R_3 \subseteq C \times D$ then $(R1 \circ R2) \circ R3 = R1 \circ (R2 \circ R3)$ I don't know where exactly to ...
1
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1answer
22 views

Trig and Inverse Trig Function Compositions

Sorry if this is a dumb question, but I honestly tried searching and all I could find was obvious stuff like $\sin(\arcsin(x)) = x$ So what is the logic behind simplifying expressions like this, ...
1
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1answer
19 views

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 ...
2
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0answers
55 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 ...
0
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1answer
56 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.
0
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1answer
82 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 ...
0
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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 ...
0
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6answers
833 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 ...
2
votes
1answer
96 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
30 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 ...
0
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3answers
46 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, ...
2
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2answers
72 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 ...
0
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2answers
29 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
64 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 ...
2
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0answers
38 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 ...
0
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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
34 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
35 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 ...
0
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4answers
83 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: ...
0
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1answer
50 views

Write the function as a combination of elementary functions

$2xe ^{(-4x^2)}$ Is this correct? $f(x) = -4x^2, g(x) = e^x, h(x) = 2x$ $h(x)\cdot g(f(x))$
1
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1answer
37 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 ...
2
votes
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
50 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. ...
1
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2answers
89 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 ...
0
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2answers
101 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 ...
1
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3answers
76 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 ...
0
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3answers
36 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 ...
0
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2answers
89 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 ...
0
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1answer
37 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 ...
0
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2answers
641 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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0answers
22 views

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"?
0
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1answer
66 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.
0
votes
1answer
25 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
56 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 ...
4
votes
2answers
1k 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 ...
1
vote
1answer
27 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 ...
2
votes
2answers
176 views

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

$A=\{1,2,3,4,5\}$ How many functions $f : A \to A$ so that $f$ is onto? $5!$ is this correct? How many functions $f : A \to A$ so that $(f\circ f)(1) = 3$? $f(1)=1, f(1)=2, ...
7
votes
3answers
173 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 $f=g \circ g$.