Elementary questions about functions, notation, properties, and operations such as function composition.

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1answer
19 views

Generating function for any series

Given a summation series, is there any way to generate a function to find the value of the sum of first n terms? For example, we have, $\sum f(n) = f(0) + f(1) + ... + f(n)$ . Now, I want to know ...
0
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0answers
23 views

Show that if $f$ is surjective then there is a function $g:Y\to X$ such that $f\circ g$ is the identity function on $Y$.

Let $X,Y$ be nonempty sets and $f:X\to Y$ be a function. Show that if $f$ is surjective then there is a function $g:Y\to X$ such that $f\circ g$ is the identity function on $Y$. My approach: If $f$ ...
1
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1answer
17 views

Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one.

Let $X,Y$ be nonempty sets and $f:X\to Y$ be a function. Show that if there is a function $g:Y\to X$ such that $g\circ f$ is the identity function on $X$, then $f$ is one to one. My approach is by ...
10
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1answer
83 views

Does $f(x) \in \mathbb{Z}[x]$ irreducible, imply $f(2x)$ also irreducible?

It is well known that $f(x+a) \in \mathbb{Z}[x]$ is irreducible when $f(x)$ is irreducible. I was wondering whether $f(2x)$ and in general whether $f(nx)$ is irreducible as well. Though it is ...
3
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0answers
31 views

Is there a name for this property of multiplication (and other functions)?

Suppose $x,y \in \mathbb{R_+}, x<y$, and $ 0 < \varepsilon \leq (y-x)/2$. It seems to me that $xy < (x+\varepsilon)(y-\varepsilon)$ and equivalently that $(x+\varepsilon)(y-\varepsilon)$ is ...
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2answers
78 views

Is it legal to define a function like this?

So I need to do something recursively and count how many steps it takes and I've come up with something like this: $$ f(x)= \begin{cases} f(\operatorname{Re}(x)+1)+i, & ...
2
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3answers
29 views

Can distributions be thought of as functions of a real variable?

I understand that, given some function space, distributions lie in the dual space. In that sense, they can be thought of as functions of a "function of a real variable" variable. But the common ...
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3answers
47 views

Tricky Integration And Functions Question

If there is a functions $f(x)$ such that $$ f(x) = x+\int_0^{\frac{\pi}{2}} \sin(x+y)\cdot f(y) \, dy $$ I tried doing it but it seems to get more and more complex as I proceed. Find $f(x)$ Thanks
2
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1answer
42 views

Correctness of proof that weak convergence implies pointwise convergence in C([0,1])

I want to prove that in the space of (complex-valued) continuous functions on the real interval [0,1] equipped with the sup norm, which I will denote by $\mathscr{C}([0,1])$, weak convergence implies ...
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1answer
21 views

Randomly picking 2 integers to compute a third one with equiprobability

I have a problem, that might be simple but I just don't see it for the moment. Supposing you have a finite set of integers $S_1$, I am looking for a simple function that when randomly picking two ...
2
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0answers
56 views

Find the value of $f(0)$, where $F'(a)+2$ is the area bounded by…

Question: Let $$F(x)=\int_{x}^{x^2+\frac{\pi}{6}}2\cos^2tdt$$ for all $x\ \epsilon \ \mathbb {R}$ and $f:[0,\frac12]\to[0,\infty)$ be a continuous function. For $a \ \epsilon \ [0,\frac12],$ if ...
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1answer
18 views

vertical line test for a function

The function f is defined by $f:\mathbb{R}\to \mathbb{R},\ f(x) = 3x-4$ For what value of $k$ is $f(k)+f(2k)=0$? How do I find k? I tried substituting $f$ for $4/3$ but got the wrong answer I also ...
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0answers
23 views

Composition of functions, injectivity, and surjectivity [on hold]

(a)Prove or disprove: For any sets X,Y,Z and any maps f:X→Y and g:Y→Z, if f is injective and g is surjective, then g ◦ f is surjective. (b)Prove or disprove: For any setsX,Y,Z and any maps f:X→Y and ...
0
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1answer
13 views

Investigation of function with parameters

I'm not so sure how to find the parameters "a" and "b" of this function, and ill be happy if someone could help me: $$y=\frac{ax+bx+1}{x^2-6x-8}$$ The slope of the tangent function is: ...
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0answers
11 views

Single-statement Continuous Periodic function without trigonometry and complex numbers

Superseding the question Periodic function without trigonometry and complex numbers , I am now asking: Can I get a single-statement continuous periodic function without using trigonometric functions ...
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0answers
15 views

If the composite function $f_1(f_2(\cdots(f_n(x))\cdots)$ is increasing and $r$ number of $f_i$s are decreasing while rest are increasing…

If the composite function $f_1(f_2(f_3(\cdots(f_n(x))\cdots)$ is an increasing function and if $r$ number of functions are decreasing functions while the rest are increasing, find the maximum value of ...
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0answers
20 views

find the stationary points for $f(x)=x^{\frac 2 3}$.difference between the stationary point and critical point and one more called turning point.

Find the stationary points for $f(x)=x^{\frac 2 3}$. My work I realized the following $\spadesuit$ $f'(x)=\frac 2 3 x^{-\frac 1 3}$ which is not defined at $x=0$ $\spadesuit$ $f'(x)<0$ for ...
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2answers
24 views

Injective functions have no functional extrema

Is it true that all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that are injective (one-to-one) have no functional extrema? I can't find any counterexample, but that can easily be proved when we ...
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1answer
15 views

how to write a function in terms of Heaviside step function

I'm reading Paul Online Notes. There's an example of writing a function in terms of Heaviside step function as follows: $$ f(t) = \begin{cases} -4 &\text{if } t < 6, \\ 25 &\text{if } 6 \le ...
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1answer
31 views

$g\left( x \right)=3-\sqrt { x+3 } $

To be honest, I suck at math, so that's why I'm here, to ask you guys a question. The function is $g\left( x \right)=3-\sqrt { x+3 } $. When the value of $x$ is equal to $-3y$ will equal $3$. If $x$ ...
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2answers
36 views

Is it true that $|f(x)|\leq |f^2(x)|$?

Is the following true for all $x\in\mathbb{R}$ and for all real functions f? $$\left| f(x)\right| \leq \left| f^2(x)\right|$$ Also, is it true that $|f(x)|\leq |f^3(x)|$?
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2answers
22 views

The supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ when $x+y=2n$ for some fixed $n\in \mathbb N $

Let $S$ be the set of all tuples $(x,y)$ such that $x+y=2n$ for a fixed $n\in \mathbb N$. Then what is the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ $?$ I substituted $y=2n-x$ ...
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1answer
31 views

How do we call a pair of sets $A,B$ such that there is some injection $f: A \to B$?

Let $A,B$ be sets and let $f: A \to B$. If $f$ is a surjection, then we may simply write $f(A) = B$ or say in a more laborious way that $f$ maps $A$ onto $B$, to mean the same thing. However, if $f$ ...
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3answers
45 views

Give a good reason to define a function from A to B as a triple (F, A, B) rather than a functional set of pairs with domain A and image included in B.

The operative part of this question is "good reason": either an example or an argument, without preconceptions or fallacies. The object is comparing two definitions for "a function f from A to B", ...
4
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1answer
25 views

Minimum value function

It's just a very simple question, is there a function defined and that tells you the minumum and maximum value of a list of variables, like: min(4, 3) = 3 min(2, 19) = 2 max(1, 10, 3) = 10 Is that the ...
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2answers
32 views

What is the meaning of Right Hand Limit at $\infty$?

For a limit to exist, the left hand limit must equal the right hand limit. That is, $$\lim_{x\to c^+}f(x)=\lim_{x\to c^-} f(x)$$ However, if $x\to\infty$, then what does the right hand limit mean ? ...
5
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3answers
102 views

Range of a Rational Function

How to find the Range of function $$f(x)= \frac{x^2-3x-4}{x^2 - 3x +4}$$ I tried to equate the expression to $y$, then cross multiplied $$ y= \frac{x^2-3x-4}{x^2 - 3x +4}$$ $$ y(x^2 - 3x +4)= ...
4
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2answers
219 views

How do we call a pair of sets between which there is a bijection that need not have additional property?

Let $A,B$ be sets and let $f: A \to B$. Then we say that $A,B$ are isomorphic under $f$ if $f$ is a linear function that maps $A$ onto $B$ in a one-to-one manner; that $A,B$ are homeomorphic under $f$ ...
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2answers
40 views

Intuitive explanation of the Dirichlet function and rationality

The Dirichlet function is defined by $f(x)=\begin{cases} c &\text{ if } x\in \mathbb{Q}\\d &\text{ if } x\notin \mathbb{Q}.\end{cases}, c\neq d$ See MathWorld's page for the full definition. ...
3
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0answers
20 views

Characterize in terms of fibre

I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you. For a function $f:X\to Y$, and y an element of ...
2
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2answers
36 views

What function is this? $\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$ [on hold]

I am interested to know what function represents the following series: $$\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$$
4
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1answer
41 views

A continuous bounded function from $\mathbb R$ to $\mathbb R$ can be increasing or not?

Let $f:\mathbb R \rightarrow \mathbb R$ be a continuous and bounded function , then $a$) $f$ has a fixed point. $b$) $f$ cannot be increasing $c$) $\lim_{x\rightarrow \infty} f(x)$ exists. ...
0
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1answer
27 views

How do you symbolically represent the general principle of induction? [on hold]

Normally a specific function is given, and then it would be asked to prove the validity of that specific function with induction. But how do you logically represent the general principle of induction ...
0
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2answers
63 views

Periodic function without trigonometry and complex numbers [on hold]

Can I get a periodic function without using trigonometric functions or complex numbers? UPDATE: The question has been superseded by Single-statement Continuous Periodic function without trigonometry ...
3
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2answers
66 views

Are all operations functions?

I have looked at Wikipedia(I know it's not completely reliable) but on it an operation is formally defined as: "A function ω is a function of the form $ω : V → Y$, where $V ⊂ X_1 × … × X_k$." and I ...
2
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2answers
27 views

Fibers of an Element

Prepping myself for a graduate abstract course and we are using Dummit and Foote's Text. We are starting on Chapter 1, so I thought it would be a good idea to go over chapter 0. There is a term that ...
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3answers
50 views

Injective function $g:B \to A$ from a surjective function $f:A \to B$

I wish to prove the existence of an injective function $g:B\to A$ given a surjective function $f:A\to B$. This sounds simple enough, however I'm having trouble writing a formal proof for it. Thanks ...
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0answers
25 views

decomposing a function into embedding and projection

I have a simple question. If $f:\mathbb{S}^{2}\rightarrow\mathbb{R}$ is a non-constant continuous function, can we represent it as a composition $f=p\varphi$, where ...
0
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1answer
18 views

Limit of a Monotonic Increasing and Non-Bounded Function

I have made a solution for the following question and I'm wondering if it's correct. I think that something is missing here. Can you help me complete the solution? Let $f$ be a function. The ...
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1answer
16 views

About the stable/invariant point sets in a plane with respect to shift/linear transformation

I'm reading Vlademir A. Zorich's Mathmatical Analysis I, meeting exercise question as following: a) A set $S \subset X$ is stable with respect to a mapping $f:X \rightarrow X$ if $f(S) \subset ...
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3answers
29 views

Equality of One-One functions. [on hold]

We're given two functions $f$ and $g$ , both $f$ and $g$ are one-one and onto. Now, if we say that $f(x) = g(x)$ for a value of $x$, can we conclude that $f=g$ ?
3
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2answers
28 views

Minimum of $f(x)=\sum_{i=1}^n\frac{a_n}{x-b_n}$ occurs at extreme point?

Let $a_1,\ldots,a_n$ be real numbers and $b_1,\ldots,b_n>1$. Define $$f(x)=\sum_{i=1}^n\frac{a_i}{x-b_i}.$$ Is it always true that $f(x)\geq\min\{f(0),f(1)\}$ for all $x\in[0,1]$?
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1answer
32 views

Does anyone know of an entire function $f$ such that $f(x,y)=0\iff\exists n\in\Bbb{N}\left[y=nx\right]$?

I tried constructing $f$ as $f(x,y) = n\left(\frac{y}{x} \right)$ where $n(a)=\lim \limits_{x \to a} \frac{\sin(\pi x)}{x}$, but I can't figure out how to remove the discontinuities when $x$ or $y$ is ...
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1answer
14 views

Classify the growth of functions and find a more general growth function

The following function $f(t,x):[0,T]\times R\mapsto R$ such that $\int^T_0|f(t,0)|^2 d t<\infty$, where $0<T<\infty$. If $f(t,x)$ satisfies $|f(t,x)|\leq Ax+B$ for each $x\in R$ and $A, B$ ...
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0answers
23 views

Restriction over pdf such that an integral inequality holds $\int_{-\infty}^{+\infty}\left(F(x)-\frac{2}{3}\right)xf(x)dx\geq 0$

Let $f(x)$ be a pdf in $(-\infty,+\infty)$ and $F(x)$ it's cdf. Assume both are smooth. I need to find restrictions over the pdf such that the following inequality holds: ...
0
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1answer
49 views

How many distinct roots $ax^5+bx^3+cx+d$ has

$a,b,c>0$ How many distinct roots $ax^5+bx^3+cx+d=0$ has? question doesnt clarify which kind of root it has. and I dont understand why the question didnt say 'may has' . because by ...
0
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0answers
19 views

Can I get anywhere with working out an unknown function if input is restricted? [on hold]

I want to know the formula that a game is using for some task (doesn't really matter what exactly). I know what the inputs and outputs are, but there's a finite number/combination of inputs - I know ...
0
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0answers
22 views

How to obtain accumulated counts of past events by time $t$?

Given $f: [0, \infty) \to \{0,1\}$, $f(t)$ represents whether there is an event occurring at time $t$. How can we obtain $g: [0,\infty) \to \mathbb{N}_0$ so that $g(t)$ represents the number of ...
10
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4answers
213 views

Find all functions f such that $f(f(x))=f(x)+x$

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(f(x))=f(x)+x, \forall x\in\mathbb{R}$. Find all such functions $f$. Clearly, $f$ is an "one-to-one function". I have tried setting ...
0
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1answer
15 views

Find maxima and minima of the function

Given: $$f:\mathbb{R}^2 \rightarrow \mathbb{R}, f\left(x,y \right)=-x^4+x^3-3x^2y+3xy^2-y^3$$ Find all points where gradient is equal to zero. Decide whether in those points function has either maxima ...