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

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0
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1answer
51 views

Is there any equation for this type of skewed parabola?

I am having the following parabola looking curve (blue curve), but its not exactly symmetric. The best fit that I am getting using a quadratic equation is also shown (black curve) but it is not ...
0
votes
0answers
29 views

Is there a complete function for the integers?

Completeness is used to describe many things, but in logic it is often used to describe a set of operators from which any input-output mapping can be created. Does there exist a function which can ...
1
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0answers
25 views

Question about composed functions [closed]

Let X be a finite set and $f : X → X$ a function. Let $f^1 = f$ and if $f^ n$ has been defined for n ∈ N then set $f^{n+1} = f ◦ f^ n$ . a) Prove for some n, $f^{n+1} = f^{n}$. b)Set Y = $Range(f^n)$ ...
0
votes
0answers
48 views

Proof about functions

Let $f : X \rightarrow Y$ be a function, $B$, $C$ subsets of $Y$. If $B \cap C = \emptyset$, prove that $f^{-1} (B) \cap f^{-1} (C) = \emptyset$. Approach suppose $x\in f^{-1} (B) ∩ f^{-1} (C) \...
1
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2answers
26 views

What is a saturated function?

I couldn't find a definition online. I know that the sigmoid function is saturated but what does it mean.
4
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0answers
78 views

Why is the unit circle definition of trig functions not rigorous enough?

It has recently come to my attention that the usual unit circle derivation of the elementary trigonometric functions isn't considered rigorous enough. Apparently, this has to do with problems ...
0
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0answers
17 views

How can I see that the image of an open subset of an open set under a function is an open set?

Let A an open set of ℝⁿ and ƒ:A→ℝᵐ of class C'(A). If for some P₀∈A, ƒ'(P₀) is surjective, prove that exists δ>0 such that B$_{δ}$(P₀)⊂A and for all open subset Ω⊂B$_{δ}$(P₀) it holds that ƒ(Ω) is ...
1
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1answer
20 views

The biggest positive integer which divides $f(n) - n\;\;\forall n \in \mathbb{N}$

Let $f: \mathbb{N} -\mathbb{N}$ be a function defined by $f(x)=$biggest positive integer obtained by reshuffling the digits of $x$ . For example $f(296)=962$ Question: Find the biggest positive ...
0
votes
1answer
36 views

Can a function which is multiplied by “e” have a maximum?

Short question, Lets say I have a function $f(x,y,z) = e^x * (\ldots)$. So, no matter what is written in the brackets, the function won't ever have a maximum point, right? Or are there certain ...
5
votes
3answers
158 views

Find the Taylor series of $f^{\circ n}=\underset{n\text{ times}}{\underbrace{f\circ f \circ f\ldots \circ f}}$

Define $f(x)=ln(1+x)$. Then $f^{\circ 2}(x)=ln(1+ln(1+x))$, and $f^{\circ 3}(x)=ln(1+ln(1+ln(1+x)))$, etc. Find the Taylor series of $f^{\circ n}=\underset{n\text{ times}}{\underbrace{f\circ f \circ ...
7
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0answers
85 views

Let $f$ be a twice differential function on the open interval $(-1,1)$ such that $f(0)=1$. Show that $f'(0) ≥ -\sqrt2$ [duplicate]

Q. Let $f$ be a twice differential function on the open interval $(-1,1)$ such that $f(0) = 1$. Suppose $f$ also satisfies $f(x) ≥ 0$, $f'(x) ≤ 0$ and $f″(x) ≤ f(x)$, for all $x ≥ 0$. Show that $f'(0) ...
0
votes
0answers
8 views

proving the output of cantor pairing function is un correlated if one bit of input is changed

Pairing function $\pi :N \times N\rightarrow N$ is defined as: $\pi(a_{1},a_{2})=\frac{1}{2}(a_{1}+a_{2})(a_{1}+a_{2}+1)+a_{2}$. Exp 1. If $a_{1},a_{2}$ are input then output is $\pi_{1}$ Exp 2. If ...
1
vote
1answer
246 views

Prove that continuous functions mapping irrationals to rationals must be constant

Let $f\colon[0,1] \to \mathbb{R}$ be a continuous function such that any irrational number is mapped to a rational number. Then $f$ must be a constant. Well, the context isn't that much, I was ...
1
vote
1answer
66 views

Books about the foundations of (calculus) functions?

I'm looking for a foundational book that builds up ideas like transcendental functions. For example, how the trigonometric functions are truly defined when plotted as continuous functions. I believe ...
2
votes
1answer
29 views

Lexicographic index of strings with repeated characters

As the title says, I need to find a way (or a function, specifically) to convert a string, let's say $AABACCAB$, to it's lexicographic number among its permutations. The number of different words ...
2
votes
2answers
53 views

find $f\circ f(x)$ for this funtion

Find the $( f\circ f)(x)$, if $ f(x) = \begin{cases} 1+x,\,\,\,\,\,\,\,\,\,\,\ 0\leq x\leq2 \\ 3-x,\,\,\,\,\,\,\,\,\,\,\, 2<x\leq3 \end{cases}$ My attempt: $$ f(f(x))=\begin{cases}1+(1+x),\,\...
1
vote
2answers
57 views

For $f(x)=\frac {1}{x}$ and $g(x)=\sqrt{x-4}$, find the domain of the composite function $g\circ f(x)$.

For $f(x)=\frac {1}{x}$ and $g(x)=\sqrt{x-4}$, find the domain of the composite function $g\circ f(x)$. My Attempt Here, $$f(x)=\frac {1}{x}$$ $$g(x)=\sqrt{x-4}$$ Now, $$g\circ f(x)=\sqrt{\frac {1-...
0
votes
1answer
18 views

lemma about absolutely continuous function

i found this Lemma about absolutely continuous function Let $u [0,T] \rightarrow \mathbb{R}^d $ an absolutely continous function , then \begin{equation} \int_0^T <\dot{u}(t), u(t) > dt = \...
0
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0answers
19 views

If the function $\rho(r)$ is monotonically decreasing can we prove that T(d) will have an inflexion point?

This is a function that comes up when analysing the behaviour of oscillators in radially symmetrical fields \begin{equation} {\mathbf{T}} (d) = \int_{- \sqrt{{R}^2 - d^2}}^{ \sqrt{{R}^2 - d^2}} {\frac{...
0
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2answers
53 views

Does such kind of function exsists? [closed]

Consider the signum function denoted by $f$. Does there exist a function $g$ such that $g′(x) = f(x)$ ?
1
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2answers
33 views

How do I calculate gradient?

$q(x)=x^TAx+b^Tx+c$ Where A is matrix, $x,b\in \mathbb{R}^n $ $c\in \mathbb{R}$ So someone in my book wrote that q(x) is the same like $q(x)=a_{11}x_1^2+...a_{nn}x_n^2+2a_{12}x_1x_2...+2a_{ij}...
1
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2answers
37 views

What is a totally defined partial recursive function?

Alright, so I've always thought that a partial function was a function from $A$ to $B$ whose domain is only a subset of $A$. A total function, on the other hand, I took to be a function whose domain ...
1
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2answers
47 views

How to define a function recursively?

I'm currently trying to work out a function where f(x) = 2x + 4 The question is to define f(x) recursively I'm unsure what this means, I have an idea of how to do it but I'm not sure if it's right. ...
1
vote
1answer
39 views

Prove $\Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor$ , n is a Natural Number

Prove the following identity: $$\lfloor x\rfloor +\lfloor x+\frac{1}{n}\rfloor +\lfloor x+\frac{2}{n}\rfloor +\lfloor x+\frac{3}{n}\rfloor+...+\lfloor x+\frac{n-1}{n}\rfloor =\lfloor nx\rfloor$$ ...
2
votes
0answers
61 views

What is this function, and what are it's properties?

I made a function that determines how "prime-y" a number is; if $f(x) = 1$ then $x \in primes $. The function is $$f(x) = \frac{\pi(x) - \#\{p \in primes | p<x, p \space| \space x\}}{\pi(x)}$$ ...
1
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3answers
40 views

Concerning the Countability of the Set of Reals with Decimal Representations Consisting of All $1s$

The Problem: Exhibit a one-to-one correspondence between the set of of positive integers and the set, $S$, of real numbers with decimal representations consisting of all $1$s. Where I Am: I realize ...
1
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0answers
21 views

Are these correct extension of injective and surjective functions?

Recall the definition of injective and surjective functions: 1.a A function $f: X \to Y$, is injective if for all $a,b \in X$, if $a \neq b$, then $f(a) \neq f(b)$ 1.b A function $f: X \to Y$,...
0
votes
0answers
10 views

Which areas or curves has function when it has one constant and two constants?

$ f: \left[0,\infty \right)\times \left[0,2\pi\right] \times \left[-\frac{\pi}{2}, \frac{\pi}{2} \right] \mapsto \mathbb R ^{3}\begin{pmatrix} r \\ \phi \\ v \end{pmatrix} \mapsto \begin{pmatrix} r \...
3
votes
2answers
29 views

removable discontinuity

I am having some confusions about the removable discontinuities. most of the functions with removable discontinuity I have faced are piece wise functions.example: $f(x)=\begin{cases}x+2,x\neq 1 \\4,x=...
-2
votes
2answers
37 views

What is the graph of the function $y = \frac{x}{1+ |x|}$?

In this question , multiple concepts of graphical transformations are involved. I am facing problems in applying all of them in a single question.
2
votes
0answers
23 views

Continuous function but iterated integrals are not equal

Given a sequence $(g_n)$ of continuous functions $g_n:\mathbb R \to [0,\infty)$ with the properties $\operatorname{supp} g_n^{(1)} \subset (n,n+1)$ and $\int g_n \, d\lambda=1$ for all $n$, how can I ...
1
vote
1answer
44 views

Interval for the solutions of $\{x+1\}<x^2-2x$ where $\{x\}$ is the fractional part of $x$.

Find the interval(s) which contain solutions of $$\{x+1\}<x^2-2x$$ where $\{x\}$ is the fractional part of $x$. I was told that one way of solving this would be graphically. However I generally ...
1
vote
1answer
33 views

Bijective function from $\Bbb N ^{n \times m}$ to $\Bbb N$?

I'm working on a hash function and I was wondering if there is a function which can tansform a matrix into a natural being bijective. For example: $A=\begin{bmatrix} 0&0&1\\ 2&0&1\\ 0&...
0
votes
3answers
45 views

Check for an onto function

Why is $y= x^{2006} + x^{-2006} +5$ not an onto function if $f(x):\mathbb{R} \rightarrow \mathbb{R}$ Please provide an explanation.
3
votes
3answers
62 views

Solution set for $\lfloor x\rfloor\{x\}=1$ [closed]

What is the solution set for $\lfloor x\rfloor\{x\}=1$ , where $\{x\}$ and $\lfloor x\rfloor$ are respectively fractional part and greatest integer function of $x$. P.S.: the answer is $\{m+1/m:m\in\...
0
votes
1answer
26 views

Evaluating $\frac{1}{f} \circ g$

This may be a very easy question, suppose $f$ and $g$ are some arbitrary simple polynomial functions. When trying to evaluate $\frac{1}{f} \circ g$ ... $$\frac{1}{f} \circ g = f^{-1} \circ g$$ Which ...
0
votes
1answer
35 views

If $f$ is injective, then $f(X\backslash A) = f(X) \backslash f(A)$

Given $f:X \to Y$ injective, $A \subseteq X$, then $f(X\backslash A) = f(X) \backslash f(A)$ I have spent a long time looking at this problem but I have not found a good way to approach this. Here ...
0
votes
1answer
20 views

How to convert a function to the form y = A sin(Bx + C) + D to find the phase shift, period, and frequencies?

The question asks us to find the period of the function $y=\sin(√2x) + \sin(3√2x)$. I usually know how to find the period and all that in the format of $y = A \sin(Bx + C) + D$, but how do I get this ...
0
votes
1answer
30 views

Why would the projection function open and not closed?

Given $p: \mathbb{R} \times \mathbb{R} \to \mathbb{R}, (x,y) \mapsto x$ Why is $p$ open but not closed? Shouldn't it be the case $p$ is closed not open because it sends a singleton to singleton, and ...
1
vote
3answers
66 views

Express the function $f(x)= (1-\sin x)/(1+\sin x)$ as the sum of an even and odd function. [closed]

Express the function $f(x)= (1-\sin x)/(1+\sin x)$ as the sum of an even and odd function.
0
votes
2answers
41 views

Find a function that is continuous in usual topology, discontinuous in lower limit topology

I had found a function that was continuous in the lower limit topology but not the usual topology Show the Heaviside step function is continuous in $(\mathbb{R}, \mathcal{T}_\text{lower limit})$ $$h(...
0
votes
0answers
28 views

How to show that $f(X \backslash f^{-1}(Y \backslash C)) \subseteq C$

Let $f: X \to Y$ be a continuous function, and that $C \subset Y$, then claim: $f(X \backslash f^{-1}(Y \backslash C)) \subseteq C$ Attempt: Immediately we run into a problem following: ...
1
vote
1answer
104 views

How to minimize $|Ax+By + C|$ given that $x \geq 0$ and $y\geq 0$ [duplicate]

I am trying to solve problem related to absolute value function, i.e given $Z(x,y) = |Ax + By + C|$ , what is the minimum value of $Z$, if $x \geq 0$ and $y\geq 0$ and x,y belongs to integers
1
vote
1answer
49 views

Prove that either $f(x)=0$ or $f(x)$ is an odd function.

Let $f(x)$ be a derivable function at $x = 0$ and $$f(\frac{x+y}{k})=\frac{f(x)+f(y)}{k} $$ $k\in \mathbb R\setminus \{0,2\}$. Prove that either $f(x)=0$ or $f(x)$ is an odd function. Could someone ...
0
votes
1answer
21 views

Range of two equal functions

I know two functions $f$ and $g$ are equal if (1) their domains are equal (2) their co-domains are equal (3) $f(x) = g(x)$ I want to ask if two functions are equal is it necessary that they will ...
4
votes
1answer
40 views

Injection from $\mathcal P \left({\mathbb{N}}\right)$ to derangements of $\mathbb{N}$

Let $S$ be the set of the permutations without fixed points of $\mathbb{N}$. Is there an elegant way to exhibit an injection from the power set $\mathcal P \left({\mathbb{N}}\right)$ to $S$ ? (...
2
votes
2answers
22 views

Determining the image of a function $\psi:\mathbb{R}^2 \rightarrow \mathbb{R}^2$

Determining the image of a function $\psi:\mathbb{R}^2 \rightarrow \mathbb{R}^2$, $\psi(x,y) = (x^2 - y^2, x^2 + y^2)$ I made some observations about $\psi$: $\psi$ isn't injective, since $\psi(-x,-...
1
vote
0answers
25 views

How to construct continuous functions on arbitrary topological spaces?

Constructing continuous functions $\mathbb{R} \to \mathbb{R}$ (with the euclidean topology) is easy: there is a quite large collection of elementary functions, and we can get even more by composing/...
0
votes
1answer
59 views

How we create our own $\pi$ finder formula/function?

1) Nilakantha Somayaji; $\pi=3+\dfrac{4}{3^3-3}-\dfrac{4}{5^3-5}+\dfrac{4}{7^3-7}-\dfrac{4}{9^3-9}+.....$ 2)Franciscus Vieta; $\pi=2.\dfrac{2}{\sqrt2}.\dfrac{2}{\sqrt{2+\sqrt2}}.\dfrac{2}{\sqrt{2+\...
1
vote
2answers
38 views

Transformation of graph of $f(x)$ to $\frac{1}{f(x)}$

With the help of the graph given below i.e. $y = f(x)$ plot the graph of :- $1.$ $y=\frac{1}{f(x)}$ $2.$ $y=2f(x)$ $3.$ $y=f(2x)$ For $y=2f(x)$, I multiplied each coordinate of $y$-axis by 2 and ...