functions

1. functions, relations, injective,...

Hello there, I'm new to the kind of math you have to prove, so be patient ^^ Are there any improvements I could make here? In the following, I will talk about $Z$ as a ring. More specifically I consider $(Z,+,\times)$ a commutative and unitary ring, where $Z = \{...,-2,-1,0,1,2,...\}$. I...
2. composition functions

Hi everyone, can someone please look at the image that I uploaded and explain to me why they decided to multiply x^2-9 by both sides? I am a little lost here. Thanks in advance.
3. Limes superior and inferior inequalities with functions and its derivatives

Hi, I have got some troubles with Lemma 2. from N.H. Duu, On The Existence of Bounded Solutions for Lotka- Volterra Equations, Acta Mathematica Vietnamica 25(2) (2000), 145-159. Can anybody help me understand the last part of this proof? Exactly, how this transformation using Cauchy theorem...
4. operations with functions

Hi, it's a math problem. I can't remember the rules for adding, subtracting, multiplying and dividing fractions with roots. For example, f(x) the square root of x+1 added to f (g) 2/the square root of x+1. The answer was x+3 over the square root of x+1, but I don't know how they obtained this...
5. Generalization of Riemann-weil formula for arithmetic functions

\sum_{n=1}^{\infty} \frac{\mu(n)}{\sqrt{n}}g(\log n)=\sum_{\gamma}\frac{h( \gamma)}{\zeta '( \rho )}+\sum_{n=1}^{\infty} \frac{1}{\zeta ' (-2n)} \int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x} [/itex] \sum_{n=1}^{\infty} \frac{\lambda(n)}{\sqrt{n}}g(\log n) = \sum_{\gamma}\frac{h( \gamma)\zeta(2...
6. Which of the following statements is correct ?

This is a question from a college admission exam. There is a mistake in the attachment. f is derivative not differentiable.
7. Functions

Hello all, Determine the functions f:\mathbb R \rightarrow \mathbb R that verify the relationship f(x)=f^2(\{x\})-f([x])+1 , \forall x\in \mathbb R and where \{x\} and [x] are defined as in the following site: Home Math Notes Algebra II TRIGONOMETRY INTEGER PART OF NUMBERS. FRACTIONAL...
8. How to find functions & inputs whose output is a specific number

I'm interested in the following problem: given a random number n (n can be gigantic), how do we find a pair function+input(s) whose output is n such that the input(s) are relatively small in size? This problems arises in data compression; consider the bits that make up a file (or a substring...
9. Functions

Hello all, Determine ascending functions f:\mathbb N^{*} \rightarrow \mathbb N^{*} with the property that \frac{f(1)+f(2)+\cdots +f(n)}{f(1)f(2)+f(2)f(3)+\cdots +f(n)f(n+1)}=\frac{3}{2f(n)+4} \forall n \in \mathbb N^{*}. All the best, Integrator
10. Functions in function - synthesis of a general form

Hi All, I've one problem related to defining general form of fuction f(x) which is constructed by using functions inside of it. Let we say, that f(x) is defined as follows: f(x)= \frac{c \cdot a_{0}+ f_{1}(x) }{ a_{0}+ c \cdot f_{1}(x)} and the two next functions are: f_{1}(x)= \frac{c...
11. Functions in function - synthesis of a general form

Hi All, I've one question related to a function, which can be defined by using other functions. Let we say, that such situation can be defined as follows: f_{0}(q)= \frac{a_{0}+c \cdot f_{1}(q)}{c \cdot a_{0}+ f_{1}(q)} f_{1}(q)= \frac{a_{1}+c \cdot f_{2}(q)}{c \cdot a_{1}+ f_{2}(q)}...
12. Compare the functions

Compare A_n with B_n for n\in \mathbb{N} A_n = \frac{1}{e^{-1^2 }+e^{-2^2 }+e^{-3^2}+...+e^{-n^2 }} . B_n =(1+e^{-1})\cdot (1+e^{-2}) \cdot ... \cdot (1+e^{-n}). In short-terms : A_n=(\sum_{\lambda=1}^{n} e^{-\lambda^2 })^{-1}\; \; and \; \; B_n=\prod_{\lambda=1}^{n} (1+e^{-\lambda }) .

In this exploration, you have the chance to use what you have learned about trigonometric functions and their graphs by building a function that models a real periodic situation (a merry-go-round). Watch the video â€œMerry-Go-Round.â€ (Merry-Go-Round descriptive transcript) pt. 1 Assume...
14. Linear independence of functions

Whether Cos 3x and Cos (3x + Pi/2)are linearly independent or not in the interval (-infinity, infinity)? My attempt: I calculated the Wronskian that comes out to be -3 (independent of x) that signifies that this function is linearly independent because the sufficient condition for the set...
15. Looking for something like Penner's Easing Functions, but for larger sets of data

Hey all, I work in CG, I'm a technical director (I help build tools that facilitate the work of animators in 3D animated productions, VFX and the like). We work with keyframes a lot, where we store the transforms of certain objects and parts of characters over time. These keys are represented...
16. Proving an Inequality for Locally Integrable Functions

>$\textbf{The Problem:}$ Let $f\geq 0$ be a bounded function and $E\subset\mathbb R^d$ have finite measure. Prove that there exists $R>0$ such that for all $0<r<R$ we have $$\int_{E}f(x)dx\leq 2\int_{E}\left(\frac{1}{\vert B(x,r)\vert}\int_{B(x,r)}f(y)dy\right)dx.$$ Here $B(x,r)$ denotes the...
17. generating functions

How can I continue this problem. my aim is to write a closed form. ( the first line is the original question given, the rest is I have written )
18. Source code for Mathematical functions.

Reference: https://www.programiz.com/c-programming/library-function/math.h/log Can we have the Source code of Math functions viz Log(), Sin(),Cos() etc? Thanks & Regards, Prashant S Akerkar
19. can anyone help me with? FUNCTIONS?!

If f(x) satisfies f(x+y) = f(x) + f(y) for all x,y belonging to $\mathbb{R}$, and f(1) = 5, find $\!$ m \sum [f(n)] n=1
20. Range of function

For which value of a does the interval [0,1/3] completely include the range of the function: f(x)=1/(3x^(4)âˆ’8ax^(3)+12a^(2)+x^(2)+a)