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

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0
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3answers
37 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
1
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0answers
31 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 ...
0
votes
1answer
20 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
votes
0answers
51 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 ...
0
votes
1answer
16 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 ...
-1
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0answers
21 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
votes
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: ...
0
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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 ...
0
votes
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 ...
0
votes
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 ...
1
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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 ...
1
vote
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 ...
0
votes
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 $x$ ...
0
votes
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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votes
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 $y=2n-x$ ...
1
vote
1answer
29 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 $f$ ...
0
votes
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
votes
1answer
24 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 ...
0
votes
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
votes
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
votes
2answers
216 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 $f$ ...
0
votes
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
votes
0answers
18 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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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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votes
3answers
49 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 ...
2
votes
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
votes
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 ...
1
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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 ...
-1
votes
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
votes
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 $x\in[0,1]$?
0
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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 ...
0
votes
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 $A, B$ ...
0
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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
votes
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
votes
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
votes
0answers
21 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
votes
4answers
212 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
votes
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 ...
-1
votes
2answers
25 views

Map real numbers into [0:255] using fixed limits interval

I got an interval from x0 to x1. I want all numbers inside this interval to be mapped (preferably linear) from 0 to 255. All numbers below x0 should be mapped to 0. All numbers above x1 should be ...
0
votes
5answers
62 views

Are these two expression equal?

My friend insisted that $(-1)^{(-n)}$ is equivalent to $(-1)^n$ for any number of $n$. A quick check in the Wolfram Alpha show ...
1
vote
2answers
64 views

finite vs infinite set function composition

If there is a set $X$ which is finite with $f : X \rightarrow X$ and $g: X \rightarrow X$, then $f \circ g = 1_X$ iff $g \circ f = 1_X$. How is it true for finite sets? I'm not too sure, but the ...
1
vote
1answer
31 views

Can true surjection really exist for algebraic functions?

Quoting a definition from Wikipedia: A surjective function is a function whose image is equal to its codomain. Consider an arbitrary algebraic function that has $\mathbb{R}$ as its codomain. ...
0
votes
0answers
9 views

Tweaking function to reduce the rate of decay of a logarithmic based curve

Im not even sure if this is possible or perhaps I may need to use a different function altogether but I currently have one that looks like this: $$y = a\log(x+b)+c$$ That produces the red curve ...
0
votes
0answers
10 views

trying to use function transformations with ln

I'm trying to understand if I can use function transformations in the usual sense (vertical and horizontal shift, stretching, etc...) with ln. Specifically, if I look at the graph of ln(x^2) and then ...
1
vote
4answers
52 views

Show that the function is continuous

To show that the function $f: \mathbb{R}^2 \rightarrow\mathbb{R}$ with $f=\left\{\begin{matrix} \frac{x^3-y^3}{x^2+y^2} & , (x,y) \neq (0,0)\\ 0 & , (x,y)=(0,0) \end{matrix}\right.$ is ...