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

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Global minima of multivariate constrained linear function

I have a function of form $ax+by+cz$ where $a, b, c$ are real numbers. Also $x, y, z$ are greater than equal to 0 and $x +y+z$ less than equal to $C$(constant). What's the global minimum of this ...
0
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3answers
31 views

Functions defined below are bounded or not?

I have a set of functions ${{x}_{i}}:\mathbb{R}_{0}^{+}\to \mathbb{R}_{0}^{+}$ ($i=1,2,...,n$). Denote the independent variable with $t$. Suppose, there are ${{\eta }_{1}},{{\eta }_{2}},\ldots ,{{\eta ...
2
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4answers
75 views

From Spivak (Foundations)(2nd edition)

I just finished high school and started reading Spivak's calculus. I've noticed that "this kind of mathematics" is kind of different and more severe from what I've been taught in high school. ...
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1answer
64 views

Find the inverse of $xe^{1-2x^2}$, $x\geqslant1$ [closed]

Find the inverse of $xe^{1-2x^2}$, $x\geqslant1$ This question is killing me and this is my last resort. Thank you in advance I've found great help on here before.
1
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1answer
34 views

What does “p-q bijection” means?

I know what 1-to-1 bijection means, nut I've found that annoying expression "Sometimes, one constructs a p — q bijection instead of a 1-1 bijection." And I truly don't understand what this supposed to ...
1
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1answer
56 views

Consider the increasing, concave function $x^{0.5}$ on $[0, 1]$.

Consider the increasing, concave function: $$ g(x) = \sqrt x, x ∈ [0, 1]. $$ Can you state a continuous function: $$ f(x), x ∈ [0, 1] $$ such that $f(0) = 0, f(x)$ is twice continuously ...
7
votes
3answers
209 views

Show that there's no continuous function that takes each of its values $f(x)$ exactly twice.

I need to prove the following: There's no continuous function $f:[a,b]\to \mathbb{R}$ that takes each of its values $f(x)$, $x\in [a,b]$ exactly twice. First of all, I didn't understand the ...
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1answer
42 views

$f:[0,1\to\mathbb{R}]$ such that $f(0) = f(1)$. Prove that exists $x\in [0,\frac{1}{2}]$ such that $f(x) = f(x+\frac{1}{2})$

I'm reading a proof that: $f:[0,1\to\mathbb{R}]$ such that $f(0) = f(1)$. Prove that exists $x\in [0,\frac{1}{2}]$ such that $f(x) = f(x+\frac{1}{2})$ It says the follwing: Define $\phi(x) = f(x+...
1
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1answer
108 views

Unknown “formula” [closed]

Hi I have problem with finding right "formula" (Is it what it is called) I am bad at maths so forgive me. So problem is this. I can insert some number and then it calculates me answer with this "...
6
votes
4answers
86 views

$f:\mathbb{R}\to\mathbb{R}$, continuous, such that $xf(x)>0$ when $x\neq 0$. Show that $f(0)=0$

I need to prove: $f:\mathbb{R}\to\mathbb{R}$, continuous, such that $xf(x)>0$ when $x\neq 0$. Show that $f(0)=0$. Show that if we remove the continuity this result will fail. Give an example. For ...
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1answer
47 views

$f:[0,1]\to [0,1]$ continuous then exists $x_0\in [0,1]$ such that $f(x_0) = x_0$

I need to prove the following: $f:[0,1]\to [0,1]$ continuous then exists $x_0\in [0,1]$ such that $f(x_0) = x_0$ $f$ continuous, then there are $x_1,x_2$ such that $f(x_1)=0$ and $f(x_2)=1$ and by ...
0
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0answers
19 views

functional type equation

Let $f,g$ be two nonconstant positive functions on $I=[0,1]$ and we assume that : $$ \sup_{x\in I}\sqrt{f^2(x)+g^2(x)}=\sqrt{\sup_{x\in I}f^2(x)+\sup_{x\in I}g^2(x)} $$ This implies that $(f,g)$ are ...
6
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4answers
202 views

Number of functions $f\colon\{1,2,3,\dots,n\} \to \{1,-1,i,-i\}$ satisfying a certain condition

What should I do here? I don't even know where to start from. Please help me by giving me a hint. Find how many are the functions: $f: \{1,2,3,\dots,n\} \to \{1,-1,i,-i\}$, where $n \geq 2$, such ...
0
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1answer
25 views

Existence of additive non-linear function

The following question should have a positive answer: it is taken from Example 1.11 of the book "Positive Operators" by Aliprantis and Burkinshaw. Question: Does there exist an additive function $\...
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1answer
27 views

Risk seeking utility

I am stuck on a question in an archived course on BerkeleyX's CS188x Artificial Intelligence. Which of the following would be a utility function for a risk-seeking preference? That is, for which ...
0
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0answers
22 views

Inverted pi/sum function

Have seen a strange symbol on a math keyboard app. It looks like the product function, but upside down: It even has space for subscript and superscript, just like $\sum$ and $\prod$: I think it's ...
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1answer
76 views

Show that $ \lim_{x\to 0} \left( \cos(\sin(x)) + \tfrac12x^2\right)^ {\left[(e^{x^2}-1)\left(1+2x-\sqrt{1+4x+2x^2} \right)\right]^{-1}} = e^{5/24}$ [closed]

I can't find following limit: $$ \lim_{x\to 0} \left( \cos(\sin(x)) + \tfrac12x^2\right)^ {\frac{1}{(e^{x^2}-1)\left(1+2x-\sqrt{1+4x+2x^2} \right)}} = e^{5/24}$$ I've tried l'hospital's rule, and ...
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0answers
20 views

If $|a+b+p+q|=\frac{k}{18}$, then find the value of $k$

Let $$f(x)= \begin{cases} ax(x-1)+b & x<1 \\ x+2 & 1\leq x\leq 3 \\ px^2+qx+2 & x>3 \end{cases} $$ be continuous for all x except $x=1$ but $|f(x)|$ is ...
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2answers
33 views

examples of first strictly concave then convex function?

I want to find out a continuous function on $[0,L]$, $L$ is a positive number, which looks like this red curve: The function is always positive. This function firstly is strictly concave, then ...
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2answers
52 views

Why are absolute values involved in functions of random variables?

From a textbook: If $X$ is a continuous random variable, then so too is the new random variable $Y = Y (X)$. The probability that $Y$ lies in the range $y$ to $y + dy$ is given by $$g(y)=\int_{...
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2answers
44 views

$\forall \epsilon>0, \exists \delta>0 / |x-a|<\epsilon\implies |f(x)-L|<\delta$

I'm asked to analyze what happens when I have $\delta$ exchanged with $\epsilon$ in the limit definition like this: $\forall \epsilon>0, \exists \delta>0 / |x-a|<\epsilon\implies |f(x)-L|<...
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1answer
20 views

weight/window functions with constant sum for infinite discrete sampling, like triangle functions

Imagine a function $w_n(x) : R \rightarrow R , n \in N$ such that: $w_n(x) = 0 , \forall x \notin [-nL, nL], L \in R $ $ 1 = \sum\limits_{k=-\infty}^{\infty} w_n(x - kL), k \in N, \forall x \in R$ ...
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1answer
41 views

Decomposing $\ln(x)$ into sum of even and odd function.

Can somebody help me break $\ln(x)$ into sum of even and odd function. As far as I know every function can be broken in such manner. Not being able to do this as $\ln(-x)$ and $\ln(x)$ cannot exist ...
0
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0answers
16 views

Induced operation and anticommutativity

Let $\odot$ and $\circledast$ be operations on X and Y. Let $f:X\to Y$ satisfy $f(r_x)=r_y,\ f(x\circledast y)=f(x)\odot f(y),\ x,y\in X$. Prove: $\ x\sim y:\Leftrightarrow f(x\circledast y)=r_y \...
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1answer
37 views

Good Books on relations and functions [closed]

What are the books you would recommend to starters on the topics of Relations and functions. In your opinion why is this book better than the others.
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1answer
26 views

Closure of a function

"Let $f: A \rightarrow A$ and let $X \subseteq A$. Then, in a ‘top down’ version, the closure f[X] of X under f is the least subset of A that includes X and also includes f(Y) whenever it includes Y. ...
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2answers
23 views

Determining domain and range

Intro I am trying to find a systematic method of finding the domain and range of a function. If I do find a successful method, I could potentially make a computer program that calculates domain and ...
0
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2answers
33 views

What's the monotony of this function?

This is the function: $g(x) = (1+a)^x - a^x$, for some $a>0$ and $x \ge 0$ I can find the monotony for $1>a>0$ this way. Let $x_1, x_2$ be two non-negative numbers such that: $$x_1<x_2 \...
2
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0answers
45 views

A sufficient condition on a real smooth function

Let $f : [0, \infty) \to \mathbb{R}$ be a smooth function. I would like to find a sufficient condition on $f$ in order to have that $$ \liminf_{t\rightarrow \infty} \int_0^t \Big(\frac{t - s}{s} \Big)...
0
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1answer
60 views

What are all polynomials $p(x)$ such that $p(q(x))=q(p(x))$ for every polynomial $q(x)$?

I assume that $p(x)$ and $q(x)$ are both real polynomials. If I let $q(x)=c$, (a constant) then $p(q(x)) = p(c) = q(p(x)) = c\ \forall c$. So $p(x)=x\ \forall x$. Is this operation valid and how ...
0
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2answers
67 views

For what value of $c$ is $f$ periodic?

Let $f(x)=a\sin(cx)+b\cos(cx)$, where $a$, $b$ and $c$ are constants. Since $\sin$ and $\cos$ have a period of $2\pi$, if $c\in\mathbb{Z}$ then $f$ has a period of $2\pi$. How to prove the converse? ...
0
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1answer
52 views

How to find vertice by two angles and side?

I know 'alpha', 'betta', length 'c', coords: 'A' and 'B' How can i find the 'C'(coords)?
0
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4answers
78 views

How do I find the solution(s) to my second-degree equation?

$$f(x) = x^2 - 3x$$ My attempt : $$ \begin{align} x^2-3x &= 4\\ x(x-3) &= 4\\ x-3 &= 4 \\ x &= 7\\ \end{align} $$ I managed to solve one part of this problem but that one part is ...
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0answers
21 views

sufficient reason for a function to be bijective

I know of course that an application $\Phi: A \rightarrow B$ is bijective if it is injective and if it surjective. I also know that for all bijective function, there exists an inverse. My question ...
4
votes
1answer
40 views

What is the name for a function that behaves symmetrically when its arguments are scaled?

In other words, is there a name for this property of a function $f$: $$f(\alpha x_1,x_2,\ldots,x_n) = f(x_1,\alpha x_2,\ldots,x_n) = \ldots = f(x_1,x_2,\ldots,\alpha x_n)$$ Edit: I appreciate the ...
0
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2answers
25 views

If $f: A \rightarrow B$ is surjective, and $A, B$ are nonempty sets, and $X \subseteq A$, does $f(A) - f(X) = f(A - X)$?

I'm working on a proof, and the proof will be complete if this is true... but I can't find a theorem in my book that explains whether or not this is true.
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2answers
44 views

How can I prove or disprove that there exists a function such that…

Suppose we have a function $f$ of $bx-ay$ where $a$ and $b$ are two real constants, if we have for example $e^{bx-ay}$ then obviously it is a function of $bx-ay$. Can we find a function $f$ such ...
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1answer
48 views

If y is not an exterior point of $K$, then there exists a $x$ in $K$. Is it true?

For a vector $v = (x_1,\ldots,x_d)^t \in \mathbb{R}^d$, we let the function $f$ be $f(v)=|v|^2=v^tv=x_1^2+\cdots+x_d^2$. Is it possible to show that there exists a x $\in K$ which satisfies $f(x)>...
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1answer
38 views

Are there mathematical objects (like matrices) which behave like shorthand operators for complicated calculations? [closed]

Matrix multiplication involves summing a product. It is appropriate where you need to multiply things together and then add. Are there more examples like this ? Namely, to use a mathematical object, ...
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2answers
41 views

$A : \mathbb{R}^n \to \mathbb{R}^n \implies A(\mathbb{Z}^n)=\Gamma$ on theTorus

In the Analysis on Manifolds via the Laplacian page $51$, it is indicated that if $A : \mathbb{R}^n \to \mathbb{R}^n$ be so that $A(\mathbb{Z}^n)=\Gamma$, then $\text{Vol} (\mathbb{T})=\det A$. Could ...
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3answers
40 views

Is the function $\frac { x-1 }{ \ln { { \left( x \right) }^{ 2 } } } $ continuous at$ x=0$?

I would like to know whether the function shown in the title is continuous or not at $x=0$. This problem is disturbing since the function isn't defined at x=0, but the limit of the function as x ...
2
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1answer
39 views

Interpolation for $f(n),n\in\mathbb{Z}$: Does it converge?

Assume a function $f(n)$ which is defined for $n\in\mathbb{Z}$. For each period $[n,n+1]$ the function could be interpolated with a polynomial of degree $m$. The polynomials should be built in a way ...
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0answers
17 views

Polynom subspace of continuously differentiable Functions

Let $n\in \mathbb{N}$ and $a\in \mathbb{R}$. Then $\mathcal{C}^n(\mathbb{R})=:V$ and $$\langle f,g\rangle :=\sum_{k=0}^n {f^{(k)}(a)g^{(k)}(a)}$$ is a positive semidefinite Bilinear Form for all $f,g\...
0
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2answers
51 views

Meaning of Vector Space over $\mathbb{R}$ being a Subspace of $\mathbb{R^R}$

$\mathscr{P(\mathbb{R})}$ is the set of all polynomials with coefficients in $\mathbb{R}$. How are below sentences related and why? (1) $\mathscr{P(\mathbb{R})}$ is a vector space over $\mathbb{R}...
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0answers
11 views

Necessary and sufficient condition for argmax/argmin

Let $x_1,\dots,x_n$ be real variables and let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a differentiable function with a unique maximum (or mininum). Is there a necessary and sufficient condition ...
0
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1answer
13 views

Etymologies of injections and surjecteions

Why one-to-one functions are called "injections" and onto functions are called "surjections"?
-1
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3answers
68 views

Is function $F(x)= 2x^2 -3x$ increasing or decreasing [closed]

$F(x)= 2x^2 -3x$. find the range of $x$ to check whether the function the is strictly increasing and strictly decreasing.
3
votes
1answer
14 views

Function/Measure Notation in Geometric Measure Theory

I'm trying to understand a formula of this kind $$ ...=\phi_\sharp \left ( f \mathcal{H}^n \right ) $$ where $\mathcal{H}^n$ is the n-dimensional Hausdorff measure on a measure space $X$, $\phi : X ...
0
votes
2answers
28 views

What is the meaning of this theorem regarding periodic functions?

I recently got acquainted with a theorem: If $f(x)$ is a periodic function with period $P$, then $f(ax+b)$ is periodic with period $\dfrac{P}{a}$ , $a>0$. I am having a difficulty in ...
2
votes
2answers
53 views

How to calculate inverse of $y=3x+4\log(x+1)$?

How to calculate inverse of $y=3x+4 \log(x+1)$? Wolframalpha says that http://m.wolframalpha.com/input/?i=Inverse+3x%2B4+log%28x%2B1%29+&x=0&y=0