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

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Why is the CT system $y(t)=x(2t)$ invertible but its DT counterpart non-invertible?

Just for clarity, a system is invertible if distinct inputs lead to distinct outputs. That said, I have two systems, a continuous time system and a discrete time system: (1) $y(t) = x(2t)$ ...
1
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1answer
15 views

Functions that preserve equivalence relations

Quick question: Let $X$ and $Y$ bet two sets and $\sim$ an equivalence relation on $X$. I was wondering what it means to say that a function $f$: $X\to Y$ 'preserves' $\sim$ in this case. Does it ...
4
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1answer
27 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
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3answers
36 views

Help me with proof concerning functions

Problem: Let $X$ and $Y$ be non-empty sets and let $f: X \rightarrow Y$ be a function. We define $F: P(Y) \rightarrow P(X)$ by $F(B) = f^{-1}(B)$ for all $B \in P(Y)$. Proof that $F$ is injective if ...
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0answers
24 views

Proof that there is no identity to integral operation on any set of functions

The statement is: Let $f\in F, f:x\mapsto f(x)$ be a function($F$ contains sufficiently non-trivial functions). Then $\not\exists I\in F$, so that $$\int_{-\infty}^\infty If=f(0)$$ What I am implying: ...
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1answer
35 views

Find a Taylor series around $x=0$ [on hold]

I don't know how to find the Taylor series around $x=0$ for: $$f(x)=\frac{\tan(2x)-\arctan(4\sinh(x))}{\sin(x^{2})}$$ Thank you in advance.
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0answers
16 views

Convexity of matrix inverse

If $$f\colon\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}$$ where $$f(x,y)=y^T x^{-1}y$$ and dom$(f)=\{(x,y)\ |\ x+x^T\gt 0\}$, then is $f$ convex?
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0answers
28 views

The top 1% own 50% of the world's wealth - how do we turn this into a function?

This Oxfam report states that 1% of the world's richest own 50% of the wealth. But to be in the top 1% - you don't have to be a billionaire (assuming a billion is US dollar one thousand million). ...
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1answer
18 views

How find $\min_{a\in\mathbb R}f\left(a \right)$ for $f(a )=\max_{1\le k\le n}\left|x_k+ay_k\right|$?

Let them be given points in the plane $P\left(x_k,y_k\right)$, where $k\in \{1,...,n\}$. Let $f(a )=\max_{1\le k\le n}\left|x_k+ay_k\right|$ , where $a\in\mathbb R$ . How find $\min_{a\in\mathbb ...
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0answers
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Is there a name for this type of operation on graphs of functions?

Suppose I have a function $f(x)$ that is defined on $\{0\}\cup[1/2,1]$ such that $f(0)=0$ and $f(x)=1+x$ for $1/2\leq x\leq1$. I want to define the following extension of this function that is ...
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2answers
38 views

The powerset of the set of natural numbers - Cantor's Theorem

It is a fact that if $A$ is any set then there is no bijection between $A$ and its powerset $P(A)$. If $A$ is finite, this is pretty clear just by looking at the sizes of $A$ and $P(A)$. But if I ...
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3answers
62 views

Does there exist a function between arbitrary sets?

Given arbitrary sets $A$ and $B$, does there exist a function $f: A\rightarrow B$ that is injective?. Does this follow from the axioms of set theory? If yes, then which axiom?
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1answer
20 views

Linear Transformations of Functions

$\textbf{Problem}$ Define $f: \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) = mx + b$. $\textbf{a.}$ Show that $f$ is a linear transformation when $b = 0$. $\textbf{b.}$ Find a property of linear ...
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1answer
8 views

Proof concerning indexed family of sets

Let $f: A \rightarrow B$ be a function. Let $I$ be a non-empty set, and let $\left\{U_i\right\}_{i \in I}$ be a family of sets indexed by $I$ such that $U_i \subset A$ for all $ i \in I$. Proof the ...
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1answer
42 views

Program to find closest function to fit arbitrary data

I've wanted this for years, but have never come across anything; a program for Windows to find the closest function to fit arbitrary data. The data I feed it is simple: A table with two columns ...
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2answers
27 views

The range of $\frac{2^x-1}{2^x+1}$

I am trying to find the range of the function $\frac{2^x-1}{2^x+1}$. If we draw using a graph plotter we can see that the range is $-2<y<2$. To find the upper bound, I tried ...
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1answer
12 views

Composed of non differentiable functions

It will be possible to find a function $f:\mathbb{R}\rightarrow \mathbb{R}$ non-differentiable at zero such that $f\circ g$ is differentiable at zero where $g:\mathbb{R}\rightarrow \mathbb{R}$ is ...
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1answer
31 views

Is $f(x,y)=-xy$ neither concave nor convex?

Is $f(x,y)=-xy$ neither concave nor convex? I used the definition for first differentiable functions and determined it depends on the choice of points, hence it is neither.
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1answer
29 views

three elementary problems on limits of several variable . [on hold]

I'm learning limits of several variable new. Can anyone help me? Computing the following limits: $\lim_{(x,y)\to(0,0)}|x|^y$ $\lim_{(x,y)\to(0,0)}\sin(x/y)$ $\lim_{(x,y)\to(0,0)}x^2\cdot ...
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4answers
89 views

Find the inverse function $x + \sqrt {x}$

$ Y = x + \sqrt {x} $ Hello , I want to find the inverse function of this function , I know that it's injective How to prove the $f(x) = \sqrt{x + \sqrt{x}}$ is injective. but do not know how to ...
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4answers
71 views

Question about $x\mapsto f(x)$ notation.

I'm trying to learn this notation, but I have some questions regarding its uses: Why is a "$:$" used instead of "$=$" when defining the function, e.g. $f: x\mapsto f(x)$ isntead of $f = x\mapsto ...
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2answers
31 views

Properties for functions $f:[a,b] \to \mathbb R$? [on hold]

Let $f:[a,b] \to \mathbb R$ be a function. Which of the followings are true: A) If $f(x)$ is continuous then it is bounded. B) If $f(x)$ is continuous then it is increasing. C) If ...
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2answers
93 views

Solving functional equation $f(x+y)+f(x-y)=2f(x)\cos y$?

How can I solve this functional equation, where $x,y$ are any real numbers and $f:\mathbb{R}\to \mathbb R$ is a function such that : $$f(x+y)+f(x-y)=2f(x)\cos y$$ I tried substituting $x=0$ to get ...
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2answers
25 views

Discrete math functions proof

Let $\mathbb N_{\text{even}}$ be the set of all natural even numbers, and $\mathbb N_{\text{odd}}$ be the set of all natural odd numbers, the function $f:\mathcal P(\mathbb N)\to \mathcal P(\mathbb ...
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1answer
26 views

how to solve piecewise function? [on hold]

Does anyone know how to find the domain of function g? Did $1-x$ will effect the interval of $g$? Given $$f(x)=\begin{cases}\frac{1}{2},& x\in\left[0,\frac{1}{2}\right]\\2x-\frac{1}{2},& ...
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0answers
42 views

finding the formula to a given table of values

I created a spreadsheet that i filled with values i got from a game. The values may be rounded, but they were calculated, so there has to be a formula behind. ...
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0answers
10 views

Specific utility (error) function for machine learning

I need a differentiable analog of following piecewise-defined function for machine learning application: $E=E(x,y)$ when $y=1$, $E=1/(x+1)$ when $y=-1$, $E=-1/(x-1)$ $y\in \{-1,1\}$ (two values, ...
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0answers
44 views

Real analytic function with radius of convergence 1 at non-negative integers

So, as the title states, the problem I was confronted with was to find a real-valued everywhere analytic function $$f:\mathbb{R}\to \mathbb{R}$$ s.t. at every non-negative integer, k ...
2
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3answers
24 views

Finding the formula of a function based on output

This is probably something super simple, but I can't find it in my book, and I don't even know what to search for because I don't know what to call it. I'm not looking for this specific answer, but ...
1
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1answer
41 views

Are functions infinite dimensional vectors? [on hold]

Are functions infinite dimensional vectors? There are a few sources on the internet that makes this claim, but they do not cite any sources which makes me feel like they are just using it as an ...
1
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1answer
29 views

Why is $\cos((\omega+\alpha\cos(\omega' t))t)$ the wrong model for frequency modulation?

So I was trying to program vibrato, or freqency modulation, naively using the model: $$\cos((\omega + \alpha\cos(\omega' t))t)$$ Where $\alpha \lt \omega$ and $\omega' \ll \omega$. For practical ...
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1answer
28 views

Draw $-dU(x)/dx$ for $U(x)$

It's been a little while since I've done any problems like this, but I just wanted to make sure I'm on the right track. Updated attempt:
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1answer
47 views

A question about a linear algebra proof [on hold]

If $f(x)$ is a function with domain $R$ such that for all real $a, x$ it is $f(ax) = af(x)$ then there exists a real number $b$ such that $f(x) = bx$ for all $x.$ How to prove this statement?
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0answers
29 views

How to display one to one correspondence for all bit strings not containing the bit O?

This is a problem from Discrete Mathematics and its Applications From the onset I saw that this set was countable was that you could physically count these out - 1, 11, 111, 1111 and perhaps ...
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1answer
16 views

Prove that, if $f(x)<a$ for $x\in I_1$ and $g(x)<a$ for $x\in I_2$, then $\sup I_1\geq\sup I_2$.

Let us consider two functions $$f:\mathbb R\rightarrow I$$ $$g:\mathbb R\rightarrow I$$ and I a subset of $\mathbb R$. Let $f(x)\leq g(x)$, $\forall x\in \mathbb R$. Prove that, if $$f(x)<a\in I$$ ...
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0answers
132 views

How many complex functions reduce to a given x-y function?

A 2D or x-y coordinate function has a complex analog, which is formed by replacing x with with the complex variable z. That function can then be separated into real and imaginary parts. Graphing the ...
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1answer
21 views

How to find the domain of this trig function?

f(x)=sqrt(tan(2x+π)) Allright, so i know you cannot have a number less than zero under the square root sign and that tangent cannot equal π/2+nπ. So should i try to find the domain of the tan ...
0
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1answer
49 views

When the imaginary part of a function is zero?

Let $z_k=x_k+ i y_k, x_i,y_i \in \mathbb{R}$ are the complex variables. Consider a polynomial of $z_k$ and its conjugates $f(z_1,\ldots,z_n, \bar{z}_1, \ldots,\bar{z}_n).$ Question:Is there any ...
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2answers
31 views

Another question about $x_0$ in the Taylor series

When we talk about Taylor series, we say it's around point $x_0$. It's in the Taylor series formula: $$f(x) = f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2f''(x_0)}{2} + \frac{(x-x_0)^3f'''(x_0)}{6} + + ...
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1answer
31 views

Find the max and the min of the function

I have this function: $$g(t)=t^2+\cos(2t)-\cos(t)$$ $$0\le t\le2\pi$$ I made the derivative:$$g(t)'=2t-2\sin(2t)+\sin(t)$$ And except for the obvious solution $t=0$ I'm not able to find the other.. ...
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2answers
35 views

To find inverse of function [on hold]

Given $ f(x) = \begin{cases} 2x, & \text{if $x\in[0,1]$} \\ 8 - 2x, & \text{if $x\in [2,3)$} \end{cases} $ Then how to find inverse of f ?
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2answers
50 views

Find the range of $ x-\sqrt{4-x^2}$

$Y=x-\sqrt{4-x^2}$. How to find these types of functions' range? I just know that the answer is $R=\{y\in\mathbb{R}\mid-2\sqrt{2}\leq y\leq 2\}$, but I have no idea how to find it step by step.
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1answer
25 views

Can the domain and co-domain be the same set? Is this a function?

Let $A$ denote the set of all real numbers. Let $B$ denote the same set as $A$. Let $f$ be the function that, to each number in $A$ assigns the cube of the number. Is $f$ a function?
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2answers
35 views

Functions - finding the domain

Question: Consider the function: $$f(x) = \log(2x + 1) - \log(x - 3)$$ What will be the domain of this function? I used two approaches to solve this question. Both approaches got me different ...
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1answer
25 views

Smooth saturation function

I need a function similar to $$Saturation(x)=min(max(x, -1), 1)$$ except for I need it to be smooth with no jump in its derivatives. It seems $arctan$ is not a good candidate since I need it to keep ...
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1answer
20 views

Total number of distinct solution produced by polynomial

I have a function $F(x,y) = ax + by$ where $x,y$ belongs to range $[1..10^{10}]$ and $a$ and $b$ are constants, all are integers. How many distinct values can be produced by this function, please give ...
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1answer
35 views

If the cos of 27 is 0.89, how much is the csc of 27

Hey guys for my trig class we're viewing trigonometric functions and their properties. So far I have understood but I came across this problem and can't seem to solve it: Given the approximation cos ...
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1answer
20 views

Function that maps a rational number to its numerator and denominator

Question: Is there a simple way to represent a function $f:\mathbb Q\to \mathbb Z^2$ that maps a rational number in lowest terms $r=\frac ab$ to the ordered pair of its numerator and denominator ...
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1answer
19 views
+50

Understanding relation between vector valued function and function objective in an multi objective optimization problem

I try to understand the relation between "vector-valued function" and "function objective" as used in optimization problem. I understand that objective function in a multi-objective problem can be ...
0
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1answer
29 views

What is the highest order of derivative of this function $f(x) = x^5\sin(\frac{1}{x}) $ at $x=0$?

The function is defined as $f(x) = x^5\sin(\frac{1}{x}) \quad \text{for} \quad x\neq 0 \quad $ and $f(x) = 0$ for $x=0$. I can't tell by just looking at the plot. I think there might be a theorem I ...