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

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3
votes
1answer
136 views

How to bound the maximal consecutive length in a random subset of [n] as function of n?

Let $S$ be a random subset of $[n]=\{1,2,\ldots,n\}$ chosen uniformly from $[n]$'s subsets. How can I find a function $f(n)$ s.t. for any $\varepsilon \gt 0$, $$\lim_{n \rightarrow \infty} P\left[(1- ...
5
votes
4answers
302 views

If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded?

If a function is uniformly continuous in $(a,b)$ can I say that its image is bounded? ($a$ and $b$ being finite numbers). I tried proving and disproving it. Couldn't find an example for a ...
1
vote
1answer
106 views

inverse of function

Thanks for the help! Here is the solution.. i had a problem: $$f(x)=\frac{(\sqrt x+8)}{(5-\sqrt x)}$$ i had to find the inverse, so lets begin... 1) first i write in terms of $y$ ...
0
votes
3answers
166 views

The square of a measurable function is measurable

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a measurable function. I want to show that $f^2:x\mapsto (f(x))^2$ is measurable. Apparently it can be shown using the facts that the sum of two ...
0
votes
2answers
90 views

Notation for probability of either sign

I have a function $f(k) = \pm 2^k$ with probability $1/2$ of either sign. How would I express this in a cleaner notation? I'm guessing to use the Kronecker delta somehow, but I can't put a finger on ...
4
votes
3answers
189 views

name this function

Is there a function that has these properties? Points: $f(1)=\tfrac{1}{2}$ $f(-1)=-\tfrac{1}{2}$ $f(0)=0$ Bounds: $f$ is bounded between $(-1,1)$: $\forall x\in\mathbb{R}: -1 < f(x) < 1$ ...
2
votes
2answers
167 views

Math symbol for approximation of probability distribution by arbitrary function?

I want to use a symbol between two functions; $$p\text{ (symbol) }f$$ such that $p$ is a probability function and $\text{(symbol)}$ implies: we do not have access to $p$ but we approximate it with ...
1
vote
1answer
40 views

How do I approach this signals and systems question?

The question seems very abstract to me as it doesn't really describe what it is asking for. The problem states the following, let: $$x[n]=\begin{cases} n,&\text{if n is odd};\\\\ ...
1
vote
3answers
159 views

Reference request - being rigorous about a common abuse of notation.

I've completely rewritten this question, in accordance with this advice. As a motivating example, suppose we're working in ETCS. Let $\bar{1}$ denote the canonical singleton set, and assert that by ...
0
votes
1answer
2k views

What does it mean when you say that the function is bounded?

What I figured is that it means that the function has an upper bound, however I came across this text: Here since g(x) either equal or less to f(x), |g(x) / f(x)| must be bounded right? Since the ...
2
votes
1answer
200 views

Convex function inequalities

1) Let $f:\mathbb{R}\to\mathbb{R} $ be a positive, convex, continuous function. Assume $f$ satisifes the following inequality$$f(x)f(y)\leq f(xy)$$ for all $x,y\in \mathbb{R}.$ What can we say most ...
0
votes
1answer
78 views

What is the most effective way to implement Hilbert's hotel? [duplicate]

Assuming I need to find an onto and 1-to-1 function from $(a,b)$ to $(0,1)$, well that's not a hard job. But things are getting bit more complicated when I'm asked to do the exact same but from ...
1
vote
2answers
802 views

Formula for Snake Draft pick numbers

Hello I am trying to come up with a formula to calculate the overall pick number in a snake style draft. For example in a snake draft every other round the pick order reverses. So in a 10 team league ...
1
vote
1answer
37 views

A question of formality regarding limits

Let $f$ be a differentiable function over $\mathbb{R}$, with its derivative being a continuous function. Let there be a function $g$ s.t. $\lim_{x \to 0} g(x) = 0$. Now I'm required to show that ...
1
vote
1answer
123 views

What's the difference between T(V) and ImT?

Assuming I have the following Linear transformation: $\mathbb{T}: \mathbb{V} \rightarrow \mathbb{W}$ where $\mathbb {V}$ and $\mathbb {W}$ are vector space. ...
4
votes
2answers
559 views

how can we convert sin function into continued fraction?

how can we convert sin function into continued fraction ? for example http://mathworld.wolfram.com/EulersContinuedFraction.html how can we convert sin to simmilar continued fraction ?? and what ...
2
votes
3answers
468 views

Proof that $(1+1/x)^x$ is monotonic increasing

How does one prove that $(1+\frac{1}{x})^x$ is monotonic increasing for any $x \in [1,\infty)$? Thanks a million!
3
votes
1answer
320 views

How is the following function an odd function? $S(x) = \sin x/x$, $x \neq 0$

How is the following function an odd function? $S(x) = \frac{\sin x}{x}$, $x \neq 0$ I get $$\frac{\sin(-x)}{-x} = \frac{\sin x}{x}$$ which is even right? because $S(-x) = S(x)$? So unless the ...
1
vote
1answer
843 views

How would I create a exponential ramp function from 0,0 to 1,1 with a single value to explain curvature?

I need an exponential function that will take linear input from 0,0 to 1,1 and give me back an exponential shaped curve such that changes in X near the 0 point result in small increases in Y, but each ...
1
vote
0answers
78 views

Function on equivalence relation on functions

Let $E$ be the set of all functions from a set $X$ into a set $Y$. Let $b \in X$ and let $R$ be the subset of $E \times E$ consisting of those pairs $(f,g)$ such that $f(b) = g(b)$. Prove that $R$ is ...
2
votes
1answer
164 views

Equivalence relation function

Let $f:X \to X$ be an injective function from a set $X$ into itself. Define a sequence of functions $f^0 , f^1, f^2, \dots : X \to X$ by letting $f^0 = \mathrm{id}$, $f^1 = f$ and $f^n = ...
2
votes
4answers
88 views

How to show that $ n^{2} = 4^{{\log_{2}}(n)} $?

I ran across this simple identity yesterday, but can’t seem to find a way to get from one side to the other: $$ n^{2} = 4^{{\log_{2}}(n)}. $$ Wolfram Alpha tells me that it is true, but other than ...
2
votes
1answer
398 views

Injection function proof

Suppose $f$ is an injection. Show that $f^{-1}\circ f(x)=x$ for all $x\in D(f)$ and $f\circ f^{-1}(y)=y$ for all $y$ in $R(f)$. In $f^{-1}$ it is defined as "Let $f$ be a one-one function with ...
0
votes
1answer
40 views

Elementary function and injection

I am a bit confused on the definition of a function and an injection. I can't seem to distinguish the two. The definition of a function is: if $(x_1,y_1)\in f, (x_1,y_2)\in f$ then $y_1 = y_2$, ...
2
votes
5answers
578 views

Can I really factor a constant into the $\min$ function?

Say I have $\min(5x_1,x_2)$ and I multiply the whole function by $10$, i.e. $10\min(5x_1,x_2)$. Does that simplify to $\min(50x_1,10x_1)$? In one of my classes I think my professor did this but I'm ...
0
votes
3answers
286 views

Finding a monotonically increasing function with limit 1

To polish/improve a homework answer, I am trying to find a monotonically, continuous, strictly increasing function $f$ with these properties: $f(0) = 0$ $\lim_{x \to \infty} f(x) = 1$ (I don't ...
2
votes
1answer
118 views

Reconstructing a Paragraph From Random Set of Words

Goal: Take a collection of randomly shuffled words that represent x and y coordinates on a plane, re-order them such to construct the original paragraph those words came from. Each word represents ...
-1
votes
5answers
105 views

An inverse of the function $e^x$

How can I prove that the function $L(x)=\int_1^xdt/t$ which is definte on $(0,\infty)$ is an invers of $\exp(x)$. Should I work on $L(x)\circ e^x=e^x\circ L(x)=x$. I am stuck.... Thanks.
0
votes
1answer
50 views

Simple bijection: help please

I am trying to show that if two sets $A,B$ have $n$ and $m$ distinct elements respectively then $A \times B$ has $nm$ elements. I assumed that there are bijections $f:\{1, ...,n\} \to A, k \mapsto ...
1
vote
2answers
131 views

Stretching a curve towards one general direction without changing two points in a curve

Hi, I'm trying to "stretch" the following curve $f(x) = \ln(x + 1)$ so that it appears similar to the green curve I have drawn in the above picture. The blue arrow indicates which general direction ...
0
votes
1answer
130 views

Computing functions from generating functions

I am new to generating functions but understand how to derive them from given discrete numeric functions. Is there a simple way to derive the discrete numeric function given a generating function. For ...
2
votes
1answer
96 views

Construct a generating function for the components of a sum

Let $j \in Z_+$. Set $$ a_j^{(1)}=a_j:=\sum_{i=0}^j\frac{(-1)^{j-i}}{i!6^i(2(j-i)+1)!} $$ and $a_j^{(l+1)}=\sum_{i=0}^ja_ia_{j-i}^{(l)}$. Find generating function $\sum_{j}a_jx^j$ so that allows to ...
0
votes
1answer
134 views

A discrete function and its rate of oscillation

Consider a function $y[n]= \cos[w n ]$, where $n$ is an integer. I have to prove that this signal will have highest rate of oscillation at $w = \pi$. I was thinking I can take the derivatives ...
2
votes
1answer
422 views

How can I determine the similarity of these graphs/curves?

I have 3 visually similar graphs pictured below. They have similar peak patterns that are visible to the naked eye, but I want to compare their similarity mathematically. I can sum each column to ...
0
votes
2answers
174 views

Help me to find function to this graph

I've got this function: $\frac{x^2}{x^2+(1-x)^2}$ ; it gives me this blue graph (in zero - one range): Could you help me find function to achieve graph close to red one?
0
votes
2answers
70 views

Continuous function, not sure what to do here…

The question is as follows: Let $f(x) = \begin{cases} x, & \mbox{if } x<1 \\ x^2+1, & \mbox{if } x\ge 1 \end{cases}$ Let $g$ be a function such that $fg$ is continuous at $1$, and ...
1
vote
1answer
29 views

Is the function is differentiable at $x$ or $D$?

I know that a) and b) is differentiable at the given points, would you maybe explain how I should show that ? a) $f:\mathbb{R}\rightarrow \mathbb{R},\quad x\rightarrow 0,{ \quad x }_{ 0 }=0$ b) ...
0
votes
1answer
100 views

Continuity of a function, Differentiable function

The following function is given: $$f:\mathbb{R}\rightarrow \mathbb{R}, \ x\rightarrow \begin{cases} x^2\cos{\left(\frac{1}{x}\right)} & \text{for } x \neq 0\\ 0& \text{for } x =0\end{cases}$$ ...
1
vote
3answers
64 views

Establish a bijective function

How do I establish a bijection from $[1,0]$ to $$[1,0]\times [1,0]$$ that is continuous? I have not been able to succeed. Edit Sorry, it's $[0,1]$ in all cases.
-1
votes
2answers
103 views

How to prove that $f:\left]-5,5\right[\to \left[0,1\right], \ x\mapsto e^{-x^2}$ is surjective?

How can I prove that this function is surjective and monotonic? $$f:\left]-5,5\right[\to \left[0,1\right], \ x\mapsto e^{-x^2}$$
0
votes
1answer
103 views

Is there a generally accepted name for the function $f = \{0 \text{ when } x=0, 1 \text{ when } x ≠0 \}$?

In one of my computer programming projects I have defined the following quite common function: $$ f(x) = \begin{cases} 0, & \text{ when } x = 0, \\ 1, & \text{ when } x \neq 0. ...
1
vote
1answer
51 views

Does the following relation always hold?

Given two functions $$f_1(x)=g_1(x)+h(x)$$ and $$f_2(x)=g_2(x)+h(x)$$ I know that $f_1(x)$ and $f_2(x)$ are monotone increasing. If $g_2(x)<g_3(x)<g_1(x)$, then is it true that ...
1
vote
1answer
249 views

complement of a function $f: \{2n | n\in \mathbb{N}_0 \}: n \rightarrow n+1$

i am reading a textbook here and i saw, there is notion of Complement of a function. or Negation of a function definiton, this is whow i understood but it is definitely wrong how i do it, i know. in ...
0
votes
1answer
1k views

Finding the Range of a Complex Function

I am taking Complex Analysis right now and having difficulty understanding how to find the range of complex functions, it seems that there is no standard way to do it and that each problem is ...
1
vote
1answer
545 views

Proof that a continuous function is bounded below

I have this question: Assuming the theorem that a continuous real-valued function on a closed bounded interval is bounded and attains its bounds, prove that if $f\colon\mathbb R\to\mathbb R$ ...
2
votes
2answers
76 views

Functions and inverses proof

Consider the subset $\mathbb{R} \times \mathbb{R}$ defined by $D = \{(x,y):|x|+|y| = 1\}$. Describe this set in words. Is it a function? Let $f,g$ be functions, such that $$g\circ f(x) = ...
0
votes
2answers
74 views

Multiple choice question about the domain of $f/g$ in terms of the domains of $f$ and $g$

If functions $f$ and $g$ have domains $Df$ and $Dg$ respectively, then the domain of $f / g$ is given by (A) the union of $Df$ and $Dg$ (B) the intersection of $Df$ and $Dg$ (C) the intersection of ...
2
votes
3answers
165 views

Finding range of a function

Can I get the range of the following function? $$y=\dfrac{x^2-3x+2}{x^2-5x+6}$$ I cannot isolate $x$ to get $x=f(y)$. Thanking you in advance.
1
vote
0answers
71 views

Strictly convex absolutly 1 homogeneous function

Is every strictly convex, 1-homogeneous function on $\mathbb R^d$ simply a multiple of the Euclidean norm? Update: The above is no, since any p-norm on $\mathbb R^d$ is strictly convex and ...
8
votes
1answer
396 views

To get addition formula of $\tan (x)$ via analytic methods

Assume that we only know $\tan (0)=0$ and also given the relation $\tan'(x)=1+\tan^2(x)$ about $\tan (x)$ and we do not know other $\tan (x)$ relations of trigonometry. How can I get the additon ...