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

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2 views

How to verify if this is a autocovariance function?

Is this$$γ(h) = 1(h = 0) − 0.5 · 1(|h| = 2) − 0.25 · 1(|h| = 3)$$an autocovariance function? How to check this? Is there a method one can use to check if a given function is an autocovariance ...
1
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1answer
31 views

Differents between $lnx^2$ and $ln(x^2)$ Find derivative

I have this problem Find derivative for $lnx^2$. It seems that $lnx^2 \neq ln(x^2)$ since the derivative are differents using Wolfram Alpha. I don't understand how to calculate the derivative for ...
7
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0answers
63 views

Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

QN: What functions (from non-negative integers to non-negative integers) satisfy the condition $$f(f(f(n))) = f(n+1) + 1$$ Comment: Evidently $f(n) = n+ 1$ is one solution. Equally evidently no ...
1
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1answer
16 views

nth derivative of a troublesome function

I don't know where to start on this problem. I'm trying to get the 2015th derivative(at x = 0) of f(x) = x^2 * arctan(x). Doing the derivatives one by one seems a little troublesome... What do you ...
6
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1answer
26 views

Let $A$ be any subset of $\mathbb R^{+}$ , then there exist a metric space $(X,d)$ such that $d:X \times X \to A \cup \{0\}$ is a surjection?

Let $A$ be any subset of the set of positive real numbers $\mathbb{R}_+$ ; then does there exist a metric space $(X,d)$ such that $d\colon X \times X \to A\cup\{0\}$ is a surjection ?
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2answers
38 views

Prove that $f$ is NOT surjective

Let $f: Z \times Z \to Z \times Z$ defined like this: $f(x,y) = (x+y, x-y)$ Prove that $f$ is injective, and not surjective. For injectivity I did that: Let $(a,b) \in Z\times Z$ and $(c,d) \in ...
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2answers
31 views

If $|B\times A| = 15$ ,evaluate: $|A\cap B|$

If $|B\times A| = 15$ and $|A\times B \backslash B \times B| = 12$. Evaluate: $|A\cap B|$ I tried for myself and got to the conclusion that $|A\times B \cap B \times B| = 3 $ I couldn't get by ...
1
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1answer
18 views

What are spatial functions?

I was reading Einstein's paper 'Concerning an Heuristic Point of View Toward the Emission and Transformation of Light' and read came across this segment: "While we consider the state of a body to be ...
1
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2answers
14 views

Find the general expression from the antiderivative

I am having trouble computing the original function. Question states: Let $f$ be a differentiable, positive function, such that $$f'(x)=x*f(x)$$ for all real numbers x. A) Find the general ...
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1answer
19 views

Need help understanding onto function

Let function $g$ from $V = \{1,2,3,4\}$ into V be defined by: $g(n)=3$. I'm having trouble understanding why $g$ is not onto. I understand why it is not one-to-one but, since all the $y$ in $Y$, are ...
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0answers
4 views

Order of Dilated horizontally and translated horizontally

I have a parent function $f(x) = x^2$, and $g(x) = (6[x-2]))^2$ is a transformation from $f(x)$. The question is: $g(x)$ is from $f(x)$ by Dilated horizontally by a factor of 1/6, then translated ...
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0answers
18 views

Get a function (equation) from data points?

Is there a way to get a function (equation) from data points? For example, we have this famous Google's 'Batman' function: ...
0
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0answers
24 views

How do I specify a function without a defined argument?

A function $f$ with the argument $x$ is commonly written $f_x : A\to B, x\mapsto f(x)$, or $f_x : \mathbb{R} \to \mathbb{R}, x\mapsto x^2$, but say I don't want to specify the argument, how would I ...
0
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1answer
23 views

Proving the existence of a Bijection between Cartesian Products of Sets by Induction

Prove by induction that for any sets $A_1, \ldots , A_n$, there is a bijection from $(((A_1 \times A_2) \times A_3) \times \ldots \times A_n)$ to $A_1 \times (A_2 \times ( \ldots (A_{n-1} \times A_n) ...
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2answers
86 views

Why is it that $\frac{\sin 30}{\sin 18}$ is equal to the golden ratio?

If you calculate $\frac{\sin 30}{\sin 18}$, where $18$ and $30$ are in degrees, the result is $\phi$, or alternately $\frac{1 + \sqrt{5}}{2}$. I know that these numbers add up, but is there any ...
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0answers
11 views

Find a cyclic rational function such that…

I'm looking for a function of the form $\frac{f(a,b,c)}{f(b,c,a)}$ (or close to this form, e.g. $\frac{(a+b)^2}{b^2+bc+c^2}$) which is roughly equal to $\frac{b^3-a^2-b^2-a^3-ab^2}{b^2c+a^2b+b^3}$ (I ...
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1answer
34 views

Question about Aaronson Scott Quantum Computing Since Democritus

In the chapter on sets: Equality rules: $x=x, x=y$ implies $y=x, x=y$ and $y=z$ implies $x=z$, and $x=y$ implies $f(x)=f(y)$ are all valid. where $f$ is a function. But how do we know for ...
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0answers
26 views

Analyze the variation of this function $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$ w.r.t. $x$

Please, I need to analyse the variation of the following function w.r.t. $x$ : $f(x)=x \exp[a+b(x-1)] \, E_1[a+b (x-1)]$, where $E_1[a+b (x-1)]$ is the exponential integral, $b>a$, $a>0$, ...
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1answer
35 views

Functions : Injective, surjective or bijection? [on hold]

I have been asked a question in one of my test. Question : Consider the relation R is a subset of X * Y where X = [a, b] and Y = [c, d] defined by R = {(x,y): x^2 + y^2 = 1}. For each of the ...
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1answer
15 views

Generalizing 1D function to higher dimensions

I have a function in 1D given by $f(x) = \tanh(x-x_1) + \tanh(x-x_2)$. I want to generalize this to two dimensions, such that it describes a circle. The function $f(x,y)$ has to have a form such that ...
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2answers
23 views

Help with proof about functions and subsets

Problem: let $f: A \rightarrow B$. Prove that $f$ is injective if and only if for all $D \subset A$ we have that $f^{-1}(f(D)) = D$. Proof: => Suppose $f$ is injective. Let $x \in f^{-1}(f(D))$. ...
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1answer
27 views

How to tell if a function is continuous at (0,0)

I have to decide if the following function is continuous at (0,0). it's f(x,y) = x^2 + y^2 if (x,y) does not = 0, and f(x,y) = 2 if (x,y) = (0,0) so for the first one, I assume it is continuous, ...
0
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1answer
20 views

Probability generating function and a discrete random variable

A discrete random variable $X$ has probability generating function $G_X(t)$. If $Y=aX+b$ show that the probability generating function of $Y$ is given by $G_X(t)=t^bG_X(t^a)$. Hence prove that ...
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2answers
99 views

Proving a function is onto?

Let $f: \mathbb{R}\setminus \{3\} \to \mathbb{R}\setminus \{1\}$ be defined by $f(x)=\dfrac{x+3}{x-3}$ Prove that $f$ is onto: Okay, here is the deal. I just started my first abstract algebra ...
0
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0answers
25 views

Continuous and additive function is linear [duplicate]

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuous function and $f(x+y)=f(x)+f(y)$, show that $f(x)=kx$, $k\in \mathbb{R}$. I tried to define $g(x)=f(x)-kx$ and $g(0)=0 $ but don't know how to ...
0
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1answer
10 views

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) $y(t) = x(2t)$ ...
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1answer
26 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 ...
5
votes
1answer
31 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 ...
0
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3answers
39 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 ...
0
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0answers
25 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: ...
0
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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
21 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?
0
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0answers
32 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). ...
2
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1answer
19 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
4 views

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
39 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
64 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
21 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 ...
0
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1answer
9 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
66 views
+50

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
14 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
32 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
31 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 ...
2
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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 ...
4
votes
4answers
75 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 ...
0
votes
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 ...
4
votes
2answers
98 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 ...
1
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2answers
29 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 ...
0
votes
1answer
27 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},& ...