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

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137 views

Name of “inverse distribution” axiom?

We may speak of a function distributing over another, e.g., $$ f(a,g(b,c)) = g(f(a,b),f(a,c)) \qquad a \cdot (b + c) = a \cdot b + a\cdot c$$ Some logical operators have similar distribution rules, ...
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2answers
107 views

Is there some nomenclature to get the integer value of a fraction?

There is some math nomenclature to represent the integer value of a fraction? Say, $$x \in \mathbb{R},\, \textbf{foo}(x) = \text{integer part of }x$$ Then $$x = 1.823,\, \textbf{foo}(x) = 1$$
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2answers
236 views

Is there a non-constant smooth function mapping $\mathbb{R}$ into $\mathbb{Q}$

I cannot think of a non-constant smooth function which maps all real numbers into rational numbers. Can anyone give a simple example ? The simpler, the better !
3
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3answers
619 views

X,Y are independent exponentially distributed then what is the distribution of X/(X+Y)

Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together. The exercise goes as ...
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1answer
102 views

Recursively enumerable properties

In my textbook are three interesting properties listed (which I would like to prove) (1) A is recursively enumerable iff A is the domain of a partial computable function (2) A is recursievly ...
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2answers
80 views

A problem from <<Thinking in Problems>> by Roytvarf, Birkhauser

I got a problem, which turned to be from the book "Thinking in Problems How Mathematicians Find Creative Solutions" by Roytvarf, Chapter One, Jacobi Identities and Related Combinatorial Formulas : ...
2
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3answers
211 views

one to one and onto problem

Consinder the function $$f\colon\mathbb N\to \mathbb N\text{ via }f(n)=2n-1.$$ Is function one to one? Is function onto? First of all, I think it is one to one because when i put $n=1$, I get ...
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1answer
237 views

Product of an increasing and a decreasing function

I am interested on finding conditions on a function $f(x)$, $x>0$, such that $$g(x)=\dfrac{f(x)}{x^3},$$ is a decreasing function. Unfortunately, the function $f$ in my context is not in closed ...
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2answers
139 views

Injection $f:[0,1]\to [0,1]{\bf\big\backslash}\left\{ 1/n : n\in \mathbb{N}^*\right\}$ with $f(0)=0$ and $f$ continuous at $0$?

Does there exist a function $f:[0,1]\to [0,1]\setminus\left\{\frac 1n, n\in \mathbb{N}^*\right\}$ so that $f$ is 1-1, $f(0)=0$ and $f$ continuous at $0$? Does it have a closed form? I know that as ...
3
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1answer
59 views

Can a sequence of a function with a single variable be thought about as a function with two variables?

Long title, but first off is it logically ok to think of $\{f_n(x)\}$ as $f(n,x)$ where $n$ is restricted to a natural number? Second, would this at all be useful? Thus far in my study of sequences ...
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1answer
77 views

$f: \mathbb{Z}\rightarrow\mathbb{Z}$ and $f(x)= x^2-1$. Is $f$ one-to-one? onto?

Define $f: \mathbb{Z}\rightarrow\mathbb{Z}$ by $f(x)= x^2-1$. Then $f$ is a one to one function. I think it is false because if you put $3$, then $9-1=8$ which you can get it by $2\times 4$ and ...
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2answers
88 views

What's the term for a value x that satisfies the constraint $f(x) = f$ for a function f?

I know that $x$ is called the fixed point of a function $f$ if it satisfies the constraint $f(x) = x$. However, for a function $f$ if there exists some value $x$ such that $f(x) = f$ then what is the ...
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4answers
122 views

A smooth function instead of a piecewise function

I want to find a smooth function approximating f(x) as best as possible: \begin{equation*} f(x) = \begin{cases} x & \text{if } x \le a,\\ a & \text{if } x > a. \end{cases} \end{equation*} ...
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3answers
99 views

Definition of functions [duplicate]

A function is a type of relation. That is, a function $f$ from $X$ to $Y$ is a subset of $X \times Y$ where for each $x \in X$, there is exactly one $y \in Y$ such that $(x, y) \in f$. Suppose $Y ...
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4answers
76 views

A function $f$ is increasing on the closed internal $[a,b]$

Let the function $f$ be increasing on the closed internal $[a,b]$. If $a \le f(a)$ and $f(b)\le b$, prove that: $\exists x_0\in [a,b]$, such that $f(x_0)=x_0$. Thanks for your help. Note that ...
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2answers
55 views

Why do we restrict the range of the inverse trig functions?

I understand why we restrict the domain, but why do we restrict the range? Why do we necessarily care so much for the inverse trig relations to be functions? Thanks!
2
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1answer
47 views

Identifying the constants and variables in statements

Please help me to identify the constants and variables in these statements. Thanks in advance. Ratio of the circumference of any circle to its diameter. Height of a boy on a given day. Height ...
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3answers
2k views

The sum of two periodic functions need not be a periodic function

Let $f(x)=x-[x]$ and $g(x)=\tan x$. How could we see that $f(x)-g(x)$ is not a periodic function? This will show that the sum of two periodic functions need not be a periodic function. I hope ...
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1answer
64 views

Is constructing a function that DNE a sufficient counterexample to show the function does not diverge to $\infty$?

Prove or disprove: If $f(x)\to 0$ as $x\to a^+$ and $g(x)\geq 1$ for all $x\in \mathbb{R}$, then $g(x)/f(x)\to\infty$ as $x\to a^+$. Counterexample: Let $f(x)=0$ and $g(x)=1$ for all ...
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1answer
648 views

Prove or disprove: if $f$ and $fg$ are continuous then $g$ is continuous.

Prove of provide a counterexample: Suppose that $f$ and $g$ are defined and finite valued on an open interval $I$ which contains $a$, that $f$ is continuous at $a$, and that $f(a)\neq 0$. Then $g$ is ...
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2answers
95 views

Recursive function into non-recursive

I have to express $a_n$ in terms of $n$. How do I convert this recursive function into a non-recursive one? Is there any methodology to follow in order to do the same with any recursively defined ...
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1answer
34 views

Can this equation be factored down to be smaller? $\frac{(\frac{1}{p})^{n}-(\frac{1}{p})^{((n+1)-\frac{y+1}{x+1})}}{(\frac{1}{p})^{n}-1}$

I finally worked out the formula I've been looking for: $$\frac{(\frac{1}{p})^{n}-(\frac{1}{p})^{((n+1)-\frac{y+1}{x+1})}}{(\frac{1}{p})^{n}-1}$$ Is there any way to factor this down to a smaller ...
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2answers
423 views

Find all $f(x)$ if $f(1-x)=f(x)+1-2x$?

To find one solution I assumed that $f$ is even and rewrote this as $f(x-1)-f(x)+2x=1.$ By just thinking about a solution, I was able to conclude that $f(x)=x^2$ is a solution. However, I am sure that ...
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1answer
67 views

function lifting on $S^1 \times S^1$

Let $f:S^1 \times S^1 \to S^1 \times S^1$ a continuous function and $p:\mathbb{R}^2 \to S^1 \times S^1: (t,s) \mapsto (e^{2\pi i t},e^{2\pi i s})$ a covering map. if $F: \mathbb R ^2 \to \mathbb R ^2 ...
3
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1answer
89 views

Is partition function increasing function?

I have some exercises which require knowing the number of partitions of particular numbers, so I used some python code which I found on internet to compute the values of the partition function for the ...
2
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1answer
75 views

Is there a name for the function $(1 - e^{ct})/(1 - e^{c})$?

$$f(t) = \frac{1 - e^{ct}}{1 - e^{c}}$$ This is a function which is somehow a streched exponential which is zero at $t = 0$, and one at $t = 1$, where $c$ determines the curvature (with $c = 0$, it ...
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2answers
102 views

Two equal functions on a topological space

Can anybody help please help me, I have to answer this problem in topology: "Let $f$ and $g$ be continuous functions from the topological space $T$ into $\mathbb{R}$, with the usual topology. Show ...
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1answer
56 views

Group theory, function from group to another

I have two groups A) 0 1 2 3 4 5 6 A B C D E F G B) 0 1 2 3 4 5 6 G A B C D E F What is the function to map from A to B? so that ...
2
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1answer
44 views

determining sign of function containing logarithm.

I would like to know the sign of the following term in general. I tried approximately and it was negative. Is there any $m_0$ such that for all $n>m>m_0$, the following function is positive or ...
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2answers
61 views

Can this equation be factored down? $\frac{(2^{y}-2) - 2^{y-x}}{2^{y}-1} $

Can this equation be factored down so as to be smaller? Or is this as small as it will go? $$ \frac{(2^{y}-2) - 2^{y-x}}{2^{y}-1} $$
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1answer
51 views

Why is the product of $y=2,x=0$ not related to $x=2,y=0$ in this equation?

In this equation: $$ 1 - \frac{2}{\displaystyle 2^{\frac{y+1}{x+1}}} $$ (The really small bit in the power is $\dfrac{y+1}{x+1}$) I get this table: ...
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2answers
106 views

A question on an unbounded function

Does there exist a function $f$ such that it has a finite value for each point $x$ of $[0,1]$, however for any nbhd of $x$ $f$ is unbounded? Thanks for your help.
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0answers
100 views

Why are complete lattice homomorphisms defined that way?

Given a function $f : X \rightarrow Y$ and a set $A \subseteq X$, there's at least two possible ways of interpreting the direct image $f(A),$ as explained here. In the notation of that question, write ...
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1answer
131 views

How to express exclusive intersections?

I have a function $f:\mathbb{R}^2 \rightarrow\mathbb{R}, \ \mathbf{x} \mapsto \sum_{i = 1}^{n} w_i \mathbf{1}_{A_i}(\mathbf{x})$ for $w_i \in \mathbb{R}$ and the $A_i$s are allowed to intersect. Thus ...
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1answer
40 views

Semi-continuity

Show that function $J$ is semi-continuous at point $v\in U$ iff $\forall \epsilon >0$ there exists $\delta >0$ such that $\forall u\in U\cap L_\delta (v)$, $J(u)>J(v)-\epsilon $.
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0answers
15 views

Search a function hash for comparison

I'm looking for a function that do the following: If $a_1 < b_1 < c_1$ and $a_2 < b_2 < c_2$ and $a_1 < a_2$ and $b_1 < b_2$ and $c_1 < c_2$ equivalent to $f(a_1, b_1, c_1) < ...
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4answers
100 views

The sine inequality $\frac2\pi x \le \sin x \le x$ for $0<x<\frac\pi2$

There is an exercise on $\sin x$. How could I see that for any $0<x< \frac \pi 2$, $\frac 2 \pi x \le \sin x\le x$? Thanks for your help.
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2answers
379 views

Representing a real valued function as a sum of odd and even functions [duplicate]

With $f(x)$ being a real valued function we can write it as a sum of an odd function $m(x)$ and an even function $n(x)$: $f(x)=m(x)+n(x)$ Write an equation for $f(-x)$ in terms of $m(x)$ and ...
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1answer
66 views

A question on a periodic function

Let $f(x)$ be a bounded real function on $\mathbb R$ and for any $x \in \mathbb R$ $$ f(x+\frac{13} {42})+f(x)=f(x+\frac16)+f(x+\frac17) \tag1. $$ What is the fastest way to compute the period of the ...
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0answers
49 views

Mathematical Microeconomics

I don't know if this is the proper place to ask, but given that it is a question that basically involve mathematical properties of functions, I think its okay. So the question defines the following ...
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1answer
155 views

Every function is the sum of an even function and an odd function in a unique way

It is known that every function $f(x)$ defined on the interval $(-a,a)$ can be represented as the sum of an even function and an odd function. However How do you prove that this representation is ...
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1answer
36 views

A question on inverse functions

Let $f:\mathbb R \to \mathbb R$ is a strictly increasing function and $f^{-1}$ is its inverse function. It satisfies: $f(x_1)+x_1=a$; $f^{-1}(x_2)+x_2=a$. What is the value of $x_1+x_2$? Thanks ...
3
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3answers
132 views

Continuity of $f(x)=[x]+ \sqrt{x-[x]}$

Consider the function $f(x)=[x]+ \sqrt{x-[x]}, \, x\in \Bbb R$ ; where "$[ \space ]$" denotes the greatest integer function. It is obvious that if $b$ is an integer, then $$\lim_{x\to b-} ...
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0answers
46 views

Representing series $f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$ as a Dirac comb function.

Consider the function $$f(t)= \frac{\pi c^2}{l^2} \sum_{n=1}^\infty \frac{ n }{\omega_n}\cos(\omega_nt)$$ where $\omega_n= \sqrt{(\frac{n \pi c}{l})^2-(\frac{r_0}{2})^2}.$ If we neglect the term ...
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1answer
297 views

Let $f(x,y,z)$ be a convex function. Is the reciprocal convex?

Suppose that $f(x,y,z)$ be a convex function. Prove $\frac{1}{f(x,y,z)}$ is convex. Or give an example of $f$ where $1/f$ is not convex. For example, I know that $f(x,y,z)=(1+x^2)\sqrt{1+y^2+z^2}$ ...
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2answers
2k views

Is it possible to find function that contains every given point?

Let say we have a arbitrary number of given points and there is at least one function, for which every point lies on its graph. Is it possible to find that function using only X and Y coordinates of ...
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3answers
168 views

Is it possible to find out $x^2$ parabola and function from 3 given points?

I am programming a ball falling down from a cliff and bouncing back. The physics can be ignored and I want to use a simple $y = ax^2$ parabola to draw the falling ball. I have given two points, the ...
5
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1answer
103 views

Is the variant direct image mathematically significant?

Preimages have the property that for an arbitrary function $f : X \rightarrow Y$ and all $B \subseteq Y$ it holds that $$f^{-1}(B^c)=[f^{-1}(B)]^c.$$ However, the analogous statement for direct ...
6
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2answers
217 views

Is There A Function Of Constant Area?

If I take a point $(x,y)$ and multiply the coordinates $x\times y$ to find the area $(A)$ defined by the rectangle formed with the axes, then is there a function $f(x)$ so that $xy = A$, regardless of ...
1
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1answer
97 views

What is the range of the following function?

I have difficulties in understanding the concept of range. Let $f:\mathbb Z_{12}\to \mathbb Z_3$, $f(x)=x$. What is the range of it? Here is what i think: Range of $f$ is the set $\{a \mid a\equiv x ...