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

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6 views

How to prove that the following function has a unique mode?

I am trying to prove that the function $$f(\alpha)=n\ln \alpha-n\ln\Big(\sum_{i=1}^{n}t_i^\alpha+\int_{a}^{b}x^{\alpha+\beta}e^{-\lambda x^\beta}\,dx\Big)+(\alpha-1)\sum_{i=1}^{n}\ln t_i,$$ where ...
1
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2answers
26 views

Uniqueness of log function with relaxed conditions?

Question If: $$f(a) + f(b) = f(ab)$$ $$ f(1) = 0 $$ $$ a<b \implies f(a) < f(b) \forall a,b \in N $$ where $N$ is the set of natural numbers. Prove or disprove $f$ must be the $\log$ ...
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0answers
20 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta}e^{-\lambda x^\beta}\,dx$,is ...
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0answers
35 views

In which cases are $(f\circ g)(x) = (g\circ f)(x)$?

I have found three cases: 1) If $f$ and $g$ are the same function. 2) If $f$ and $g$ are mutually inverse. 3) If both are polynomials of degree $1$ Maybe there are more.
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0answers
27 views

how to get the solution?

I take the next exercise from Apostol's book. I thought for a while , but nothing. Then I read the solution. I try to think: how did he get it , but nothing. I want to know how to get to the solution. ...
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1answer
18 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
3
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2answers
27 views

Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find ...
2
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2answers
16 views

Vector-Valued Functions and Continuity

Why is it that when a vector-valued function $r(t)$ is continuous at some time $t$ then $\|r(t)\|$ is also continuous at that time $t$, but the converse is not true? That if $\|r(t)\|$ is continuous ...
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2answers
29 views

Show that an inverse of a bijective linear map is a linear map.

So I've got a bijection. It clearly has an inverse, but how exactly do I prove that the inverse is a linear map as well? enter image description here
2
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1answer
23 views

Codomains and the definition of a function

A function $f$ is defined as a set of ordered pairs $(x, y)$ such that $(x,b), (x,c) \in f \Longrightarrow b=c$. Since $y$ is determined uniquely by $x$, it is customarily denoted $f(x)$. One ...
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2answers
49 views

Find the inverse $\dfrac{x}{\|x\|}$ in $\mathbb{R^2}$

I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$ Let's do this. Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$ $$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$ Then $$y_1 = ...
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votes
1answer
20 views

How can I show the points of continuity of the following function

How can I show the points of continuity of the following function $$f(x) = \begin{cases} 2x, & \text{if $x \in \Bbb Q$} \\[2ex] x+3, & \text{if $x \in \Bbb I$ } \end{cases}$$ I am having ...
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2answers
16 views

Neural Network Sigmoid Problem

I currently try to create a three layer neural network and I want to use the backpropagation. My training data is: Input : Output 0 | 1 : 0 1 | 1 : 2 In many ...
-1
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0answers
22 views

Function Limit & Continuity [on hold]

What is Function Limit & Continuity? I'm a little bit silly.Is there anyone to explain those terms precisely? Thanks in Advance...
-3
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2answers
56 views

Help with continuity [on hold]

Could you please clarify these questions to me. Find all the numbers for which the given function is discontinuous. $F(x)=[x-1]$ I think the solution is $\Bbb Z$ numbers right ? $F(x)= ...
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1answer
14 views

If $f(r,t)=g(r),\,\forall r,t$ would that make $f$ and $g$ constant functions?

I know that if $f(r)=g(t),\,\forall r,t$ then $f(r)=g(t)=constant$, but If $f(r,t)=g(r),\,\forall r,t$, where now $f$ depends on $t$ would that lead to the same conclusion i.e. ...
-1
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0answers
36 views

Simplify $|a+sx|$ if $|x|=x$ and $|s|=1$. [on hold]

Simplify $|a+sx|$ if $|x|=x$ and $|s|=1$. Forgive me, this is probably really simple. Thanks. Edit: $a=v+sx$, where $v$ is a symmetric random variable with zero mean. So we can't say anything ...
0
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0answers
12 views

Is there a way to reverse engineer an already large number to make it smaller? [on hold]

Using the Ackermann function for example, it's quite easy to make massive numbers. My question is whether there's an existing algorithm that can take a large number and reverse engineer it to make an ...
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0answers
11 views

Let $S=[0,1) \cup [2,3]$ and $f:S \to \Bbb R$ be a strictly increasing map such that $f(S)$ is connected. Which of the following statements is true?

$f$ has exactly one discontinuity. $f$ has exactly two discontinuities. $f$ has infinitely many discontinuities. $f$ is continuous. I know theorems related to connectedness and ...
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2answers
36 views

Functions between metric spaces (and how they relate to closures of sets)

Let $(X,d)$ and $(Y , p)$ be metric spaces. Prove that if $f : X \to Y$ is continuous, then for any set $A\subset X$ with closure $\overline{A}$ we have $f(\overline{A})\subset \overline{ f(A) }$ ...
2
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0answers
65 views

How do I to solve this special type of integral

Does the following function can be simplified or solved? $$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$ where S is a ...
0
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1answer
22 views

Is tangent monotonically increasing?

According to wolfram a function is monotonic if its derivative never changes sign, but the derivative doesn't have to be continuous. So I feel the answer is Yes, tangent is monotonically increasing. ...
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2answers
30 views

Prove that there is C such that $f(c)=0$

$$f\in C[a,b] $$If given that for every $ x\in[a,b]\text{ there exists } y\in[a,b]$ such that: $$|f(y)| \le \frac{1}{2}|f(x)| $$ Prove that there exists $ c\in[a,b]$ such that : $$f(c)=0$$ What ...
1
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1answer
30 views

Decomposition of function of bounded variation

Suppose we have $f:\mathbb{R} \rightarrow \mathbb{R}$ which is of bounded variation. I would like to show that it can be presented as a sum of left and right continuous functions of bounded variation. ...
3
votes
5answers
97 views

Why does $\DeclareMathOperator{arccot}{arccot}\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$ diverge?

Why does $$\lim_{x \to 1}\arccot\left(\frac{x^2+1}{x^2-1}\right)$$ diverge? In my textbook it says that from the positive side it's zero, and from the negative side it's $\pi$. However, when entering ...
0
votes
1answer
26 views

Bound the variation of this decreasing function

Let $f(x)$ be a decreasing function defined over the interval $[0,a]$, with $f(0)=b$. The first derivative of $f(x)$, which is negative, is such that $f^\prime(x) > g(x)$, or equivalently ...
1
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1answer
11 views

Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
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0answers
16 views

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well.

Show that any two different functions in $C(\Omega)$ differ as generalized functions as well. Let $f,g \in C(\Omega)$. Since $f\ne g$, there is a $x_0 \in \Omega$ such that $f(x_0)\ne g(x_0)$. Hence ...
0
votes
1answer
29 views

Function is differentiable in all the points of its domain

I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of ...
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2answers
28 views

Asymptotes of $\arctan (2x)$

My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, ...
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1answer
19 views

Composite of functions with absolute value range

I have a really big problem with this next task, determing the range of composite g(f(x)). $f(x)=-2x+4$ $g(x)=\lbrace{x^2-4};|x| \le 2\rbrace$ $g(x)=\lbrace{4x^2-x^4};|x|>2\rbrace$ The result ...
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2answers
45 views

Minimum and maximum of a two variable function

I have to study the type of critical points of the function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$ and find minimum and maximum on the generic circle centered in $ (0,0) $ and radius $ r>1 $. I ...
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0answers
33 views

Help with Proofs (Even and Odd Functions) [on hold]

How do you prove: (My attempts below) 1.Sum of 2 even functions is even ----------F(x)=f(-x) +g(-x) ---------- = f(x) + g(x) 2.Difference of 2 odd functions is even ----------F(x)=f(-x)-g(-x) ...
2
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0answers
24 views

Writing $\mathrm{SO}(2)$ as the zero-set of a function

Here I'm assuming $M_{2 \times 2}(\mathbb{R}) \cong \mathbb{R}^{4}$. The definition of $\mathrm{SO}(2)$ is: $\mathrm{SO}(2)=\{ \ A \in M_{2 \times 2}(\mathbb{R}) \ | \ \det(A)=1 \mathrm{\ and\ ...
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0answers
6 views

Finding the revenue for a certain amount of units with a demand function.

First, I have no idea if that title makes sense at all... The problem I'm trying to figure out is asking; The price p and the quantity x sold of a certain product obey the demand equation: p=-1/9x ...
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5answers
86 views

limit of $f(x) = \lim \limits_{x \to 0} (\frac{\sin x}{x})^{1/x}$ [on hold]

Any ideas how to calculate this limit without using taylor? $$f(x) = \lim \limits_{x \to 0} \left(\frac{\sin x}{x}\right)^{1/x}$$
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votes
1answer
38 views

Piece wise function continuity [on hold]

Find all values of $a$ and $b$ so that the following function is continuous for all value of $x$. ($x\in\Bbb R$). $$ f(x)=\begin{cases}-3a+4x^5b&\text{when }x\le -1\\ ax-2b&\text{when ...
2
votes
1answer
47 views

Find all functions $F(x)$ for which $F (x) + F ((x − 1)/x) = 1 + x$

Let $F (x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equation $F (x) + F ((x − 1)/x) = 1 + x$. Find $F (x)$. This ...
2
votes
2answers
51 views

Proof on Functions /Set Theory

Let $S$ be the set of all numbers of the form $a + b\sqrt 2$ where $a$ and $b$ are rational. Let $f : S \to R$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x$ and $y$ in $S$. Then $f(x)=f(1)x$ ...
1
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1answer
39 views

Examples of holomorphic, complex differentiable, always positive functions

I am looking for classes of functions which are: 1) holomorphic 2) |f(z)|>0 for all z 3) complex differentiable (i.e. f(z)=mod(z) is not valid) ...
3
votes
2answers
32 views

Series of functions converge uniformly but sequence of functions does not

Given $a>1$ and $$f_{n}(x)=\frac{1}{1+n^{a}x^{4}}$$ I'm asked to show that for any $\delta >0$, the series of functions $\sum f_{n}(x) $ converges uniformly for $\{x \in \mathbb{R} | |x| \geq ...
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1answer
32 views

Is onto function necessarily a function?

The standard definition suggests that every element in the codomain should have a preimage. So, Can different elements in codomain or range have same domain? A worst question I think. Please reply... ...
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2answers
26 views

Banach contraction theorem exercise

Use Banach contraction theorem to find a solution of x+e^x=0 correct to 3 decimal places. Any suggestions how I should begin? Banach contraction theorem is new to me.
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2answers
50 views

How can I show that this function is discontinuous at the point $x=1$?

Suppose you had the function $$ f(x) = \; \text{ the integer part of } x $$ I wish to show that this is not continuous at the point $x=1$, which I will try to do by showing that $\lim_{x \rightarrow ...
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3answers
46 views

Derivative of given $f(x)$ at $x=0$

If given this function: $$f(x) = \begin{cases} e^x, & x \le 0 \\[2ex] -e^{-x}+2, & \text{x > 0} \end{cases} $$ How do I calculate the derivative at $x=0$? Shall I calculate by the normal ...
0
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0answers
27 views

Does there exist such a function $f(x)$ that $f(f(…(f(x))))=\left (1-\frac {1}{\sqrt[n]{x}} \right)^n?$

Let $n=11...1$ (1996 figures). Does there exist such a function $f(x)$ that for all real $x \not =0, x \not =1$ holds $$f \left ( f\left (...\left (f(x) \right) \right) \right)=\left (1-\frac ...
0
votes
0answers
17 views

Transforming parts of functions

I have a function in the form: $$ \mathrm{e}^{-t\lambda} \cdot \left[t\lambda - {(t\lambda)^2 \over 2}\right] $$ If one were to plot this for say $\lambda = \frac{2}{3}$ and $t$ from $0$ to $20$, ...
1
vote
1answer
37 views

Function with infinite maxima and minima [on hold]

Can you please give an example of a function with an infinite number of maxima and minima occurring in any finite time interval? Edit: This question came to me as I was reading on the dirichlet ...
0
votes
0answers
21 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
1
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1answer
18 views

Function Equivalent to the Maximum Operator?

All numbers are real, WLOG positive. $A + B + ... + N = T$ and $A' + B' + ... + N' = T$ I'm trying to figure out some function, f, such that if $f(A,B,... ,N) > f(A',B',...,N')$ then, ...