Tagged Questions

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 ...
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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 "...
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$f:\mathbb{R}\to\mathbb{R}$, 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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$f:[0,1]\to [0,1]$ 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 ...
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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 ...
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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 ...
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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 $\... 1answer 27 views Risk seeking utility I am stuck on a question in an archived course on BerkeleyX's CS188x Artificial Intelligence. Which of the following would be a utility function for a risk-seeking preference? That is, for which ... 0answers 22 views Inverted pi/sum function Have seen a strange symbol on a math keyboard app. It looks like the product function, but upside down: It even has space for subscript and superscript, just like$\sum$and$\prod$: I think it's ... 1answer 76 views Show that$ \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 ... 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 ... 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 ... 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_{... 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|<... 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 ... 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 ... 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 \... 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. 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. ... 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 ... 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 \... 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$fin order to have that $$\liminf_{t\rightarrow \infty} \int_0^t \Big(\frac{t - s}{s} \Big)... 1answer 60 views What are all polynomials p(x) such that p(q(x))=q(p(x)) for every polynomial q(x)? I assume that p(x) and q(x) are both real polynomials. If I let q(x)=c, (a constant) then p(q(x)) = p(c) = q(p(x)) = c\ \forall c. So p(x)=x\ \forall x. Is this operation valid and how ... 2answers 67 views For what value of c is f periodic? Let f(x)=a\sin(cx)+b\cos(cx), where a, b and c are constants. Since \sin and \cos have a period of 2\pi, if c\in\mathbb{Z} then f has a period of 2\pi. How to prove the converse? ... 1answer 52 views How to find vertice by two angles and side? I know 'alpha', 'betta', length 'c', coords: 'A' and 'B' How can i find the 'C'(coords)? 4answers 78 views How do I find the solution(s) to my second-degree equation?$$f(x) = x^2 - 3x$$My attempt :$$ \begin{align} x^2-3x &= 4\\ x(x-3) &= 4\\ x-3 &= 4 \\ x &= 7\\ \end{align} $$I managed to solve one part of this problem but that one part is ... 0answers 21 views sufficient reason for a function to be bijective I know of course that an application \Phi: A \rightarrow B is bijective if it is injective and if it surjective. I also know that for all bijective function, there exists an inverse. My question ... 1answer 40 views What is the name for a function that behaves symmetrically when its arguments are scaled? In other words, is there a name for this property of a function f:$$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 ... 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. 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 ... 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)>... 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, ... 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 ... 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 ... 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 ... 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)}$$is a positive semidefinite Bilinear Form for all f,g\... 2answers 51 views Meaning of Vector Space over \mathbb{R} being a Subspace of \mathbb{R^R} \mathscr{P(\mathbb{R})} is the set of all polynomials with coefficients in \mathbb{R}. How are below sentences related and why? (1) \mathscr{P(\mathbb{R})} is a vector space over \mathbb{R}... 0answers 11 views Necessary and sufficient condition for argmax/argmin Let x_1,\dots,x_n be real variables and let f : \mathbb{R}^n \rightarrow \mathbb{R} be a differentiable function with a unique maximum (or mininum). Is there a necessary and sufficient condition ... 1answer 13 views Etymologies of injections and surjecteions Why one-to-one functions are called "injections" and onto functions are called "surjections"? 3answers 68 views Is function F(x)= 2x^2 -3x increasing or decreasing [closed] F(x)= 2x^2 -3x. find the range of x to check whether the function the is strictly increasing and strictly decreasing. 1answer 14 views Function/Measure Notation in Geometric Measure Theory I'm trying to understand a formula of this kind$$ ...=\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 ...
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 ...
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