Functions Definition and 1000 Threads

h(x) = cf(x) + kg(x) is the linear combination of functions. What makes it linear?

If X and Y are independent gamma random variables with parameters $(\alpha,\lambda)$ and $(\beta,\lambda)$, respectively, compute the joint density of U=X+Y and $V=\frac{X}{X+Y}$ without using Jacobian transformation. Hint:The joint density function can be obtained by differentiating the...

5. Which of these quadratic functions has exactly one x -intercept? o A. y=x 2 −9 o B. y=x 2 −6x+9 o C. y=x 2 −5x+6 o D. y=x 2 +x−6 A 2. What are the x-intercepts of y=(x−2)(x+5) ? o A. (0, 2) and (0, -5) o B. (0, -2) and (0, 5) o C. (-2, 0) and (5, 0) o D. (2, 0) and (-5, 0) D 5. Which of...

Is the wave function for the positron the complex conjugate of the wave function for the electron? I've tried to google this, but I can't seem to get a definite answer from a reliable source. It seems that antimatter is derived in quantum field theory which does not concentrate on wave...

Homework Statement i have been trying to learn bessel function for some time now but to not much help firstly, i don't even understand why frobenius method works why does adding a factor of x^r help to fix the singularity problem. i saw answers on google like as not all function can be...

I'm working through the problems in the first chapter of Jackson and I'm still grappling with the interpretation of Green's functions. I understand that if I have the Poisson equation ##\nabla^2\phi(x) = \frac{-\rho (x)}{\epsilon_0}## and the Green's function ##G(x, x^\prime)## then in general...

Homework Statement A has n elements. B={0,1,2,3} {1,2,3}⊆range(f) Homework EquationsThe Attempt at a Solution So in each function we must choose those 3 numbers in the range. So let's first choose all the diffrent possiblites to choose those 3: n*(n-1)*(n-2) now for the remaining elemnts, we...

I'm having trouble finding textbook material on nonlinear functions on vectors. Just as I could define a function ##f## such that: $$f(x) = cos(x)$$ I'd like to write something like: $$f(\vec{x}) = \begin{pmatrix} f_1(x_1) \\ f_2(x_2) \\ ... \\ f_n(x_n) \end{pmatrix} $$ where ##f_i## is...

Homework Statement HiI am following this proof attached and am just stuck on the bit that says: ‘since ##\Omega## is a group it follows that ##|z-\omega|<2\epsilon ## contains..’Tbh, I have little knowledge on groups , it’s not a subject I have really studied in any of my classes-so the only...

Homework Statement Hi I am looking at this derivation of differential equation satisfied by ##\phi(z)##. To start with, I know that such a disc ##D## described in the derivation can always be found because earlier in the lecture notes we proved that their exists an ##inf=min \omega ## for...

Homework Statement Let ##R## be the ring of all continuous real-valued functions ##f : [0,1] \to \mathbb{R}## with pointwise addition and pointwise multiplication of functions as its two operations. Let ##c \in [0,1]## and denote ##M_c = \{f\in R : f(c) = 0\}##. a) Show that any ##f\in R##...

Hello! (Wave) I want to find two convex functions $f,g: \mathbb{R} \to \mathbb{R}$ such that $f(x)=g(x)$ iff $x$ is an integer.I have thought of the following two functions $f(x)=e^x$, $g(x)=1$. Then at the $\Rightarrow$ direction, we would have $f(x)=g(x) \Rightarrow e^x=1 \Rightarrow x=0 \in...

Homework Statement Determine whether or not the following sequences of real valued functions are Cauchy in L^{1}[0,1]: (a) f_{n}(x) = \begin{cases} \frac{1}{\sqrt{x}} & , \frac{1}{n+1}\leq x \leq 1 \\ 0 & , \text{ otherwise } \end{cases} (b) f_{n}(x) = \begin{cases} \frac{1}{x} & ...

Y=f(x) which passes through points: (-1,3) and (0,2) and (1,0) and (2,1) and (3,5) second function is defined: g(x)=2f(x-1) Calculate g(0) Calculate g(1) Calculate g(2) Calculate g(3)

NOTE:This is not a homework question! This is just a topic that I like very much,but don’t have the programming ability to do many of them.That’s why I post this thread. C++ is a language without built-in big integer calculation functions,so building ones that can do such job is a great way to...

Does anybody know any good books to learn math?

Homework Statement I am suppose to write a program that compares the FFT (Fast Fourier Transform Diagrams) of a sampled signal without the use of a window function and with it. The window function should be as long as the signal and the signal should have N points, N chosen as to not cause...

I'm trying to prove that the set of all square integrable functions f(x) for which ∫ab |f(x)|^2 dx is finite is a vector space. Everything but the proof of closure is trivial. To prove closure, obviously we should expand out |f(x)+g(x)|^2, which turns our integral into one of |f(x)|^2 (finite)...

I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Section 3.1: Manifolds ... I need some help in order to understand Example 3.1.3 ... ... Example 3.1.3 reads as follows:In...

$$\int x^2+3 = \frac{x^3}{3}+3x+C$$ I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions:-p

I have the statement \sin[\sin^{-1}(x)] = x \hspace{7pt} if -1 \leq x \leq 1. How can I tell if plugging in x will return x for \cos[\cos^{-1}(x)] and \tan[\tan^{-1}(x)] ? What if the positions of the regular and inverse functions were reversed? For example, \cos^{-1}[\cos(x)]. I am only...

If you would allow me to ask... if i have two convex functions , and i was to place one inside the other, i.e. convolute them...what could be said in general about the resultant function. what information about the original functions can be taken from the positions of the minima. and is there...

Definition: A function f mapping from the topological space X to the topological space Y is continuous if the inverse image of every open set in Y is an open set in X. The book I'm reading (Charles Nash: Topology and Geometry for Physicists) emphasizes that inversing this definition would not...

Homework Statement Define {x \choose n}=\frac{x(x-1)(x-2)...(x-n+1)}{n!} for positive integer n. For what values of positive integers n and m is g(x)={{{x+1} \choose n} \choose {m}}-{{{x} \choose n} \choose {m}} a factor of f(x)={{{x+1} \choose n} \choose {m}}? Homework Equations The idea...

Hello I have tried to resolve an exercise which is asking how the graph is modified according to the variables into the function. I would appreciate any help since accordin to my udnerstanding the function should increase Please, follow below: Suppose y0 is the y-coordinate of the point of...

I have a set of values and I'm trying to come up with functions to fit that data. Here is what I know about the data: It is rounded down / floored to the nearest significant digit (i.e. 1 for v1 and v3, 0.1 for v2). Columns v1 and v3 look linear (e.g. first order polynomial). Column v2 looks...

In analysis, the pasting or gluing lemma, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. Can we have a similar situation for uniform continuous functions?

Let R\subseteq A*B be a binary relation from A to B , show that R is a function if and only if R^-1(not) R \subseteq idB and Rnot aR^-1 \supseteq both hold. Remember that Ida(idB) denotes the identity relation/ Function {(a.a)|a€ A} over A ( respectively ,B) Please see the attachment ,I...

Let f be a function from a set X to a set Y, moreover, A ⊆ X. What comparison sign canput instead? to assert "f ^(−1) (f(A)) ? A" become true? (Possible signs of comparison in this : ⊆, ⊇, =. It is necessary to take into account all options. f ^(−1) - inverse of fall options.), Let f be a...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ... I need help in order to fully understand Theorem 12.7, Section 12.9 ... Theorem 12.7...

I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ...I need help in order to fully understand Theorem 12.7, Section 12.9 ...Theorem 12.7...

Homework Statement Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##. Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final...

In Theodore Shifrin's book: Multivariable Mathematics, he defines the derivative of a multivariable vector-valued function as follows: Lafontaine in his book: An Introduction to Differential Manifolds, defines the derivative of a multivariable vector-valued function slightly differently as...

Homework Statement Let R be the area in the xy-plane in the 1st quadrant which is bounded by the curves y^2+x^2 = 5, y = 2x and x = 0. (y-axis). Let T be the volume of revolution that appears when R is rotated around the Y axis. Find the volume of T. Homework EquationsThe Attempt at a Solution...

Consider the following limit where L'H Rule was correctly applied twice Determine the functions f'(x), g'(x), f(x), and g(x) needed to result in the limit given. \begin{align*}\displaystyle \lim_{x \to 0}\frac{f(x)}{g(x)} \overset{\text{L'H}}=& \lim_{x \to...

Justify the following by using table, graph and equation. use words to explain each representation f(X) = 2 x2 - 8x and g(x) = x2-3x+ 6 the points (-1,10) and (6,24)

Sorry for all the questions. Reviewing for my midterm next week. Fun fun. If someone could take a look at my proof for (a) and help me out with (b) that'd be awesome! (a) Let $\Delta$ be a partition of $[a, b]$ that is a refinement of partition $\Delta'$. For a real-value function $f$ on $[a...

Homework Statement This is a translation so sorry in advance if there are funky words in here[/B] f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ. Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...

Define $f(x)=sinx$ on $[0, 2\pi]$. Find two increasing functions h and g for which f = h−g on $[0, 2\pi]$. I know that if f is of bounded variation in $[a,b]$, it is the difference of two positive, monotonic increasing functions. However, we didn't do any examples of this in class. Is there a...

Homework Statement Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##. Homework Equations Uniform convergence: for all ##\varepsilon >...

Hi PF! The ODE $$g''(x) + (1-k^2)g(x) = f(x)\\ g(0) = y(\pi/3) = 0$$ where ##f(x)## is a forcing function and ##k \in \mathbb N## is a constant has a Green's function via variation of parameters as $$ G_L = \frac{L(y)R(x)}{W} : 0<x<y<\pi/3\\ G_R = \frac{L(x)R(y)}{W} : 0<y<x<\pi/3 $$ with...

Hello folks, I'm glad that I discovered this forum. :) You might save me. I'm hearing right now differential geometry and am having some problems with the subject. May you explain me the follwoing. We had the special case of the i-th projection. My lecturer now posited that the differential of...

Hi PF! I am trying to solve an ODE by casting it as an operator problem, say ##K[y(x)] = \lambda M[y(x)]##, where ##y## is a trial function, ##x## is the independent variable, ##\lambda## is the eigenvalue, and ##K,M## are linear differential operators. For this particular problem, it's easier...

The answer for derivative of y=5tanx+4cotx is y'=-5cscx^2. But how come on math help the answer is 5sec^2x-4csc^2x? I have a calculus test coming up and I really would appreciate if someone could explain! - - - Updated - - - Oh nvm I see my mistake!

Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...

Homework Statement f(x)= x/(1+x) What is f(f(x)) and what is its domain. 2. The attempt at a solution I found f(f(x))= x/(1+2x) and the domain: (-∞,-1/2)∪(-1/2,∞) , but it is saying that I have the wrong domain. What mistake have I made? My process for finding domain: 1. Find the domain...

Please see my attached image, which is a screenshot from Khan Academy on the limits of composite functions. I just want to check if I'm understanding this correctly, particularly for #1, which has work shown on the picture. Now my question: We are taking the limit of a composition of...

Appreciate the help needed for the attached question. Thanks!

Hi members, I have a problem with the computation of residues involve the gamma functions. (see attached Pdf file) Can you show me for the first residue with the arrows, or give a hint or a link. Thank you

Hi everyone, So I am a high school student and I am learning calculus by myself right now (pretty new to that stuff still). Currently I am working through some problems where integration leads to logarithm functions. While doing one of the exercises I noticed one thing I don't understand. I...