Functions Definition and 1000 Threads

1. a. fg(x)=2(1/2(x-1))+1 fg(x)=2(x/2-1/2)+1 fg(x)=x-1+1 fg(x)=x gf(x)=1/2((2x+1)-1) gf(x)=1/2(2x+1-1) gf(x)=x+1/2-1/2 gf(x)=x The functions functions f(x) and g(x) are inverses of each other. This can be demonstarted by f(x)=2x+1 y=2x+1 x=2y+1 x-1=2y (x-1)/2=y Thus, y=1/2(x-1) = g(x) And...

This is my attempt so far: ##0.05=\frac{30t}{200000+t}## then I solved for t. And I got 333.88 min. I feel like this is way too simple of a solution and I didn't use all of what's given in the problem. For part 2 of the problem it asks, what happens to the concentration over time. I tried to...

Hello, I know that functions can have or not asymptotes. Polynomials have none. In the case of a rational functions, if the numerator degree > denominator degree by one unit, the rational function will have a) one slant asymptote and b) NO horizontal asymptotes, c) possibly several vertical...

I've been given a curve α parametrized by t : α (t) = (cos(t), t^2, 0) How would I go about finding the euclidean coordinate functions for this curve? I know how to find euclidean coord. fns. for a vector field, but I am a bit confused here. (Sorry about the formatting)

Hello, I have attached my question and the work. I believe the answer is correct. Looking for verification. Thank you!

Hi all:) In my recent exploration of Elliptic Function, Curves and Motion I have come upon a handy equation for creating orbital motion. Essentially I construct a trigonometric function and use the max distance to foci as the boundary for my motion on the x-plane. When I plot a point rotating...

Of the various notions of convergence for sequences of functions (e.g. pointwise, uniform, convergence in distribution, etc.) which of them can describe convergence of a sequence of functions that have different domains? For example, let ##F_n(x)## be defined by ##F_n(x) = 1 + h## where ##h...

Hi everyone:) I have spend a couple of days trying to teach myself the math of orbital mechanics and have been able to generate a model of the orbital path of Haley's Comet, incorporating realistic distances and periods using Kepler's second law & ellipsoid functions. This is a GIF of the motion...

I can't help but feeling these days that I don't actually understand where most of the maths I use comes from. Unfortunately, I can't remember whether this is due to the fact that I didn't take my studies seriously until the end of undergrad, or rather that these things were never actually...

Hello, I was going to solve with a calculator the eigenvalues problem of the Schrödinger equation with Yukawa potential and I was thinking that the boundary conditions on the eigenfunctions could be the same as in the case of Coulomb potential because for r -> 0 the exponential term goes to 1...

Hey! 😊 Let $0<\alpha \in \mathbb{R}$ and $(f_n)_n$ be a sequence of functions defined on $[0, +\infty)$ by: \begin{equation*}f_n(x)=n^{\alpha}xe^{-nx}\end{equation*} - Show that $(f_n)$ converges pointwise on $[0,+\infty)$. For an integer $m>a$ we have that \begin{equation*}0 \leq...

I am reading A Course in Mathematical Analysis Volume 1 by D. J. H. Garling, and I am having trouble in the following demonstration of Section 2 Differentiation. part 4 of the test, the first part of the second inequality does not make sense, I do not understand its justification. I hoped they...

Take any trig function, say, arcsin (x). Why is the answer x when taking the inverse of sin (x)? Why does arcsin (sin x) = x? Can it be that trig functions and their inverse undo each other?

Is tan^2 (x) the same as tan(x)^2? Note: I could have used any trig function. I know that tan^2 (x) means (tan x)^2. What does tan (x)^2 mean? Is it proper notation?

The hyperbolic function ##\cosh t \text{ and } \sinh t## respectively represent the x and y coordinate of the parametric equation of the parabola ##x^2-y^2=1##. The exponential expressions of these hyperbolic functions are $$ \begin{cases} \sinh x = {e^x-e^{-x}} /2; \: x \in \mathbb R, \: f(x)...

Hi, there. I couldn't find much information about this on the net so I came here to ask if anyone here knows as I thought it would be the right forum or maybe I just wasn't looking hard enough. Please not that I don't mean skills but rather what actually works that improves brain functions like...

How many of the first 1000 positive integers can be expressed in the form $\lfloor 2x \rfloor+\lfloor 4x \rfloor+\lfloor 6x \rfloor+\lfloor 8x \rfloor$, where $x$ is a real number?

Hi All, A couple of questions, please: 1) Say df is a dataframe in Python Pandas, and I select a specific column from df: Y=df[column].values. What kind of data structure is Y? 2) I want to find the sum of two numbers: Def Sum(a=0,b=0): return a+b If I want to find a sum over sum data...

Are there any useful references or resources that intuitively show how Jacobi Elliptic functions [sn, cn, dn, etc] are geometrically interpreted from properties of ellipses? And how the Jacobi Elliptic functions and integrals can be shown to be generalizations of circular trig functions? Thanks!

f-1(f(A)) = A and f-1(f(B)) = B so options (a) and (c) are wrong. For (b), I get A ⊆ A For (d), I get B ⊆ B For (e), I get A ⊆ A So there are three correct statements? Thanks

many attemps made

kindly note that this solution is NOT my original working. The problem was solved by my colleague. I have doubts with the ##k## value found. Is it not supposed to be ##k=0.5?## as opposed to ##k=2?##. From my reading on scaling, the graph shrinks when ##k## is greater than ##1## and conversely.

I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Lemma 7.4.6 ... Lemma 7.4.6 and its proof read as follows: In the above proof by Lindstrom we read the following: " ...

I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Lemma 7.4.6 ... Lemma 7.4.6 and its proof read as follows: In the above proof by Lindstrom we read the following: " ...

I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Proposition 7.3.7 ... Proposition 7.3.7 and its proof read as follows: In the above proof by Lindstrom we read the...

I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Proposition 7.3.7 ... Proposition 7.3.7 and its proof read as follows: In the above proof by Lindstrom we read the...

Hi there, I have tried to do these questions but I don't understand. Any help would be appreciated!

Mentor note: Moved from technical section, so is missing the homework template. Im doing some older exams that my professor has provided, but I haven't got the solutions for these. Can someone help confirm that the solutions I've arrived at are correct?

judge whether the following expressions are larger than, smaller than or equal to zero i) a+b+c ii)a-b+c iii)4a+2b+c iv)c-e I know that <0, b> 0, c> 0, d> 0 and e <0 but I don't know what to do

My questions are as follows: 1. How do we find them and why do we need them? 2. What are the meanings of the mean and the median of a PDF? Are the formulae below correct? $$\int_{a}^{median} f(x) \mathrm{d}x = \int_{median}^{b} f(x) \mathrm{d}x$$ $$\int_{a}^{mean} f(x) \cdot x \mathrm{d}x =...

This is my approach: These quantities namely mass, length, and time, are all additive in nature. ##2 m + 3m = 5 m ##. If the argument of the functions mentioned in the problem statement is not dimensionless, then mass, length or time do not remain additive in the image space of the functions...

Slightly confused at what it wants me to do here?

I was solving a problem for my quantum mechanics homework, and was therefore browsing in the internet for further information. Then I stumbled upon this here: R is the rotation operator, δφ an infinitesimal angle and Ψ is the wave function. I know that it is able to rotate a curve, vector...

So far I've got the real part and imaginary part of this complex number. Assume: ##z=\sin (x+iy)##, then 1. Real part: ##\sin x \cosh y## 2. Imaginary part: ##\cos x \sinh y## If I use the absolute value formula, I got ##|z|=\sqrt{\sin^2 {x}.\cosh^2 {y}+\cos^2 {x}.\sinh^2 {y} }## How to...

I have tried to solve them. I would like to know if my answers are correct. (a) The total number of functions without any restrictions ##=n^m## The number of functions such that ##f(x)## is never ##1## ##=(n-1)^m## The number of functions such that ##f(x)=1## for at least one ##x\in S_m##...

Let's say we have a curve in 2D space that we can represent in both cartesian and polar coordinates, i.e. ##y = y(x)## and ##r = r(\theta)##. If you want the tangent at any point ##(x,y) = (a,b)## on the curve you can just do the first order Taylor expansion at that point $$y(x) = y'(a)x +...

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I know how to find integral solutions of linear equations like x+y=C or x+y+z=C where C is a constant. But I don't have any idea how to solve these type of questions.I am only able to predict that both x and y will be greater than 243554.Please help.

Here is a pic of question My attempt-: I defined functions f(x-1) and f(x+1) using f(x).After defining them,I substituted their values in the equation f(x-1)+f(x+1)=sinA. For different ranges of x,I got different equations. For 1<x<2,I got 1-x=sinA. But now I am confused.For each different...

Prove that $3^n\ge(n+3)^3$ for any natural number $n\ge6$.

This is probably a silly question, though it's confused me a little so I thought I'd ask. It is my understanding that a function is loosely defined as a mapping between two sets, whilst a variable can represent an element of either of those sets. I'll take the example of velocity, since it's...

Mod note: Member warned that some effort must be shown.

I'm looking for a document (possibly online) which describes the higher cognitive functions (such as thinking, planning, creativity, comprehension, reasoning, etc.) from the neuroscientific point of view. I found only texts of brain anatomy or other describing senso-motoric and metabolic aspects...

Suppose we have a piecewise function f(t) = exp(c*t) when 0 <= t < 2 and f(t) = 0 when t >= 2. Can the above be rewritten as f(t)= exp(at)*[H(t-0) - H(t-2)], H is a heaviside function.

Hey! :o For the functions $f:A\rightarrow B$, $g:B\rightarrow A$ and $h:B\rightarrow A$ it holds that $(g\circ f)(x)=x, \ \forall x\in A$ and $(f\circ h)(x)=x, \ \forall x\in B$. Show that it holds that $g\equiv h$. I don't really have an idea how to show that. Let $x\in A$. Then $(g\circ...

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I solve problem where I use functions which are expanded upto some order. I multiply them, make square root and make derivatives or solve differential equations. What is the right way to deal with such problem? Shall I expand all functions after every step or work with unexpanded functions and...

Hi, So I have a quick conceptual question about Parseval's theorem in this application. In previous parts of this question, we have found the average powers of both f(t) and g(t) by integration and using the complex Fourier series respectively (not sure if this is relevant to my question)...

I was wondering if there is a way to deduce the solution of the potential of a charge outside a sphere given by the image method, though Green functions. Because of a Dirichlet condition (GD(R,r')=0), I know that a solution can be written as GD=Go+L, where ∇2L=0. But in order to approach this...

I'm in a first-year grad course on statistical mechanics and something about multivariable functions that has confused me since undergrad keeps popping up, mostly in the context of thermodynamics. Any insight would be much appreciated! This is a general question, but as an example imagine...