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

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0
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0answers
6 views

transitive closure and number of elements in relation?

I see an example as follows: in relation $R=\{(a,b), (b,c), (b,d), (c,e), (d,e), (c,f), (e,a) \}$, on set $\{a,b,c,d,e,f\}$. we have $30$ elements in the transitive closure of $R$. How number of ...
-2
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1answer
46 views

Can a function map $\mathbb{R}\mapsto\mathbb{R^2}$ [on hold]

Is the mapping in the title of this question possible?
2
votes
1answer
51 views

Why does the graph of $y=\gcd \left(\frac{x}{y},xy\right)$ seem to have 4 “straight” lines?

Why does the graph of $y=\gcd \left(\frac{x}{y},xy\right)$ seem to have 4 "straight" lines? Using https://www.desmos.com/calculator for plotting.
-1
votes
2answers
50 views

Prove or disprove: If $\lim_{n\to\infty} (a_{2n} - a_n) =0$, then $a_n$ has a limit (not infinity). [duplicate]

I need to prove if this is true or false.If true then I need to Prove and if false I need to provide an example that disproves the statement.I tried many times but it didnt work. Note:- Im new so I ...
0
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0answers
13 views

How to design the analyticity of a function to find when the variable becomes zero

Suppose there is a n-arity function $f(x_0,x_1,...x_n)$ and I want to use it as fitness function. By optimizing function $f$, I want to know when variable $x_0$ or $x_1,...,$ or $x_n$ comes to zero. ...
1
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0answers
11 views

Properties Of Different Kinds Of Functions - Venn Diagram Method [on hold]

Suppose there are two sets A and B.A has m elements and B has n elements. What will be maximum number of 1)Functions 2)One-One Functions 3)Many-One Functions 4)Into Functions 5)Onto Functions ...
1
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1answer
23 views

At how many points will $\lfloor(sin x + cos x )\rfloor$ be discontinuous in the interval [0,2$\pi$]

At how many points will $\lfloor(sin x + cos x )\rfloor$ be discontinuous in the interval [0,2$\pi$] ? How should the graph be ?
-1
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3answers
57 views

How do I prove this Limit?

How do I prove $$\lim_{n\to\infty}(\sqrt[3]{a_n+1}-\sqrt[3]{a_n})=0 $$ where $a_n\to \infty$ using the standard definition of the limit, or in other words using $\epsilon$?
2
votes
1answer
21 views

Smooth function conditions

A curve defined by $x=f(t)$, $y=g(t)$ is smooth if $f′(x)$ and $g′(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
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0answers
18 views

When is it possible to bound a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with $\big|\ f(x_1,x_2,\ldots,x_n)\ \big| \le {\prod}_{i=1}^n h_i(x_i)$

Is there any result that specifies when a multivariate function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ can be bounded (either locally or globally) by a product of some functions $h_i:\mathbb{R} ...
0
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0answers
3 views

What is the local form of the function at the point of self-intersection with a contour?

I am trying to solve this question : One of the contours (i.e. loci of locations with the same value) of a generic smooth scalar function of the two-dimensional plane is roughly ...
1
vote
1answer
26 views

Time derivative of logistic function [on hold]

I was wondering whether there is a possible solution to this. If we have function $$ y_t = \frac{x_t}{1+x_t}. $$ given that $x_t>0$ we can represent it as a logistic function $$ y_t = ...
-4
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0answers
35 views

cn you help me with an answer to this question [on hold]

4\surd !\,27 - 8\surd !\,12 + \surd !\,3
0
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0answers
13 views

Combinatorics Question for generating fuctions [on hold]

Any tips/helps would be greatly appreciated! Let h_n be a number sequence where h_n = 3h_(n-1) - 2h_(n-2) with h_0 = 0 and h_1 = 1. Compute the ordinary generating function of h_n, and then compute a ...
-4
votes
1answer
29 views

Combinatorics Generating Functions [on hold]

Any tips/comments would be greatly appreciated! Compute the generating function of the number sequence $h_n = (-2)^n n^2$ where $n\geq 0$.
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2answers
28 views

Verify that $\alpha(a)\neq2$ for all $a$ where $\alpha(x): (2x + 1)/(x + 2)$

If $A= \mathbb{R} \setminus \{-2\}$ and $B = \mathbb{R} \setminus \{2\}$, let $\alpha: A \to B$ by $\alpha(x): (2x + 1)/(x + 2)$. Verify that $\alpha(a)\neq2$ for all $a \in A$. As a hint, I was ...
0
votes
1answer
23 views

Linear operators in the polynomial function space

Let's say I have the standard monomial basis for P2: {$1$, $x$, $x^2$} And I have a linear operator $T: P_{2} \rightarrow P_{2}$ defined by: $T(f)(x) = f'(x) + xf''(x) + x^2f''(x)$ Now I want to ...
0
votes
1answer
11 views

The set of values for $p \in \mathbb{R}$ for which the function$f(x)=\sqrt{\log_7(\frac{2x^2+px+5}{x^2+1})}$ is defined for every real number is:

There are five possible answers. I don't know how to solve this, I can define for when $2x^2+px+5$ is bigger than $0$ but don't know how to proceed $A:(-4,4)$ $B: (-\infty,-4)$ $C: (-\infty,4)$ $D: ...
1
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1answer
22 views

finite dimensional vector spaces of functions left invariant by translation

Let $E$ be a finite dimensional vector space of functions $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall f \in E, \forall t \in \mathbb{R}, x \mapsto f(x-t) \in E$. Example of such spaces ...
0
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0answers
27 views

Every primitive of an odd function is even (proof)

I'd like to prove that every primitive of an odd function is even. This is my reasoning; FACT: 1: if f(x) is even, then f'(x) is odd [easy to prove]; 2: if f(x) is odd, then f'(x) is even ...
1
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1answer
28 views

Meaning of $t \mapsto \phi_t(x)$

The context may well be of assistance: Consider a differential equation $x'=f(x)$. Assume that $f:\mathbb R^n\to\mathbb R^n$ is continuously differentiable. Denote by ...
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0answers
39 views

PDE first order

I'm heading the book Elements Of Partial Differential Equations -Sneddon 1957. At chapter two exists this exercise "Eliminate the arbitrary function $f$ fron the equation $$ z= ...
0
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1answer
58 views

Let $f(x) =\begin{cases} 0 & \text{if }x \leq 1 \\\ \log_2x & x > 1 \\ \end{cases}$ and let $f^{(2)}(x) =f(f(x)),f^{(3)}(x) =f(f^{(2)}(x))\ldots$ [on hold]

Problem : Let $f(x) =\begin{cases} 0 & \text{if }x \leq 1 \\ \log_2x & x > 1 \\ \end{cases}$ and let $f^{(2)}(x) =f(f(x)),f^{(3)}(x) =f(f^{(2)}(x)), \ldots$ and generally , ...
0
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0answers
18 views

Probability Density Function with Cumulative Distribution & Standard Deviation

Suppose that $X$ is a continuous random variable with probability density function given by $$f(x) = \begin{cases} a & 1 ≤ x ≤ 3 \\ 0 & \mbox{otherwise} \end{cases} $$ Then: (a) How do you ...
0
votes
1answer
53 views

Show that $f$ is one-to-one if and only if it is onto.

Suppose that $f$ is a function from A to B, where A and B are finite sets with $| A |= |B|$. Show that $f$ is one-to-one if and only if it is onto. How should I begin?
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0answers
26 views

Programming language for function analysis

what would you suggest as a programming language for working with functions and graphs. I need them to work with complex piecewise functions. P.S it is okay, if you suggest any software.
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1answer
23 views

X divided by Y, N times until a boundary is reached

not sure how to ask this but here is an example: X = 31.0 Y = 2.0 Z = 5.0 i want to keep dividing X by Y and the result of that again by Y and so on until i reach Z i will stop. assuming N is the ...
0
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0answers
17 views

Generate a function that shuffles a number withing a given range which is reproducible

Lets say I have an array of numbers $1 2 3 4 5 6 7$. I want to shuffle these numbers in some order , $7 5 4 3 1 26$ . However , it should be revesrible. That is given the second array I must be able ...
3
votes
4answers
79 views

show that $f(x)=-3x+4$ is bijective

Determine whether each of these functions is a bijection from $\mathbb{R}$ to $\mathbb{R}$ a) $f(x)=-3x+4$ So I know that a function is bijective if it is both injective (one-to-one) and surjective ...
1
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1answer
41 views

A maximization problem parametrized by a function

Let $f$ be a smooth positive monotonically increasing real function which is defined and finite in $[0,1]$, and define the following two quantities (see the figure below): $F=\int_{x=0}^1{f(x)dx}$ = ...
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2answers
41 views

2xy-y=4+5x, what rule is being applied in the next step, that leads to (2x-1)y=4+5x

So the question says it all, I am sure that's the next step, and I also confirmed it with Wolfram Alpha, I am trying to calculate the inverse of a function, but I have a memory loss of what rule is ...
0
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0answers
35 views

Is the function $ y^2=ayx+bx^2+c$ odd or even?

If we have the function $y^2=a\,y\,x+b\,x^2+c$ where $x$ is the independent variable, do we consider such function odd, or even?
1
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1answer
28 views

Logarithm multivariable limit $\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$

Find multivariable limit $$\lim_{\left( x,y \right) \rightarrow (0,0)}\frac{\ln(1+x^3+y^3)}{\sqrt{x^2+y^2}}$$ I was trying to find and inequality i've found out that: ...
1
vote
1answer
67 views

One-to-One Functions that satisfies none of the following [on hold]

Assume $A = \{1, 2, 3, 4\}$ and $B = \{1, 2, 3, 4, 5, 6\}$. How many one-to-one functions $f : A \to B$ satisfy none of the following conditions: $f(1) \in \{1, 2\}$, $f(2) = 3$, $f(3) \in \{3, 4\}$, ...
0
votes
1answer
24 views

Maximization of a function in an interval

I am writing a computer program where I have $x$ real positive varying in the domain $[\sqrt{U}, U]$. I want the value of $x$ which maximizes: $$ (1+ \sqrt{U}) - \frac{\sqrt{U}-1}{U-\sqrt{U}} x - ...
1
vote
2answers
33 views

How to estimate the axis of symmetry for an even function with error?

I have a situation here, where, for an unknown $t$, and an unknown but nice* real function $f$, for which $x\rightarrow f(x-t)$ is even, I measure $f(x) + \epsilon_x$, where $\epsilon_x$ is some kind ...
2
votes
2answers
437 views

Are all smooth functions bounded?

in my book it says that when a function f is smooth, it also means that it is bounded. I understand that a smooth function has contineous derivatives of all orders, but how can we know that the ...
1
vote
1answer
24 views

What ranges to take while finding composition functions

A function is defined as $$f(x)=\begin{cases}-x,&\quad x<0\\x,&\quad 0\le x\le 1\\2-x,&\quad x>1 \end{cases}$$ How to find composite function function $f(f(x))$? I can't ...
-2
votes
1answer
21 views

$f$ is lipchitz continuous, can I extend it to $\bar{A}$ and maintain the lipchitz continuity? [on hold]

Let $(X,d)$ be a complete metric space, $f:A\subset X \to X$ be Lipchitz continuous, does there exist an extension $\bar{f}:\bar{A} \to X$ such that $\bar{f}$ is also Lipchitz continuous?
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votes
2answers
10 views

Given two piece-wise functions, prove their composition is periodic.

I know how to graph h(f(x)), however, I am having trouble proving it is periodic. i. We must prove that h(f(x + k)) = h(f(x)) for all x ∈ ℝ and for some k ≠0 ∈ ℝ in order to prove that h(f(x)) is ...
-2
votes
0answers
24 views

meaning of $f(x)\cdot g(x) = (f\cdot g)(x)$; information lost? [on hold]

When people say $f(x)\cdot g(x)$ is the same as $(f\cdot g)(x)$, or other such statements, how do I know that normal algebraic operations that I've learned in school won't damage the information so ...
0
votes
1answer
24 views

What does it mean to 'Find the limit function $f(x) = \lim_{n\to\infty}e^{-n\; x}$?

As the question asks, I'm wondering about what it means to: Find the limit function $$f(x) = \lim_{n\to\infty}e^{-n\;x}$$ By taking the limit, I can easily see that the exponential decay function ...
2
votes
1answer
41 views

How can I compute $\frac{exp(\lambda v_j)}{\sum_{i=1}^n exp(\lambda v_i)}$ in a stable way?

Given an $n$ vector and $\lambda$ > 1e4 I wish to compute this sum $$\frac{exp(\lambda v_j)}{\sum_{i=1}^n exp(\lambda v_i)}$$ for a fixed $j \in \{1, \dots, n\}$. The sum should be less than one, ...
-2
votes
1answer
37 views

Use the Mean Value Theorem to prove an inequality [on hold]

Using MVT, prove the following equation to be true: $$\sqrt{x} - \frac{x-y}{2\sqrt{y}} < \sqrt{y} < \sqrt{x} - \frac{x-y}{2\sqrt{x}},$$ given that $y>x>0$
0
votes
1answer
23 views

Find a one-to-one correspondence (i.e, a bijection) [on hold]

Find a bijection between the following sets where {[]} denotes a closed interval and {()} denotes an open interval A = [-3,7] and B = [41,100] & A = (-∞,-3) and B = (8,∞)
0
votes
1answer
16 views

functions and recursions

The sequence s(k) where k=1,2,3.... satisfies the recursion s(n)=s(n−2)+s(n−3) for n≥4. If s(n) is rewritten in the form s(n)=s(n−1)+S(n0,n1,…) where S(n0,n1,…) is some linear combination of terms ...
0
votes
0answers
54 views

Solve $x+x^{\frac{1}{e^{-x}}}+e^{-x}=p$, where p is some real number

I started to tinker a little bit, discovered that derivative approaches 1 - surprisingly, quite quickly. If f(x) is expression on the left handside, then following inequality applies: \begin{equation} ...
0
votes
1answer
28 views

How can I write the equation of this graph?

I have this simple graph : I want to write this graph in this form : f(t)= {1, if .... and 2, if ...} I can't fill these spaces, I can't come to a solution for this no matter how I try. Can you ...
1
vote
3answers
51 views

Definite integral of function is zero

I am attempting to solve an equation wherein $\int_{-\infty}^\infty f(x) \, dx = 0$. There obviously exist some some solutions, such as $f(x) = xe^{-x^2}$ and trivially $f(x) = 0$, but is there a ...
0
votes
0answers
14 views

behavior of function

Im looking at the asymptotic behavior of the function $f(x)=x-c(\lceil \frac{x}{c} \rceil)$ as $x \rightarrow \infty$ for some constant $c>0$. I believe this function is bounded above by $0$ ...