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

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Where is $f(x)=\sqrt{|1-x^2|}$ Lipschitz continous?

It seems to me that the Lipschitz constant is 1 near $x=\pm 1$, $y= \pm 1$ $$ |f(x)-f(y)| \leq \frac{|x+y|}{\sqrt{|1-x^2|}+\sqrt{|1-y^2|}}|x-y| $$ How would you define the Lipschitz constant L?
0
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0answers
10 views

Is there a form that approximates arbitrary linear operators on continuous functions?

I was wondering if there was any operator form that was dense in the set of linear operators $B$ mapping $C(a,b) \to C(c,d)$. I thought that maybe integral transforms may be a general form; that is ...
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3answers
51 views

How do I show algebraically that the period of the tangent function is $\pi$?

How do I show that the positive real number $p$ for which $\tan (x+p)=\tan (x)$ is equal to $\pi$? In essence how do I prove the period of the tangent function is $\pi$? Please bear in mind I am a ...
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1answer
61 views

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x … $ are all integers then $x$ is an integer as well?

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
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2answers
22 views

help finding the inverse of this function

$$f(x)=x^3 - \frac4x$$ Find the value of the inverse for $x=6$. The answer is $2$ and I'm having problems finding the inverse function because there are two variables of different powers.
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0answers
16 views

Composite function of a multiple condition function.

Q. $f\left(x\right)=1+x\:for\:0\le x\le 2$ $f\left(x\right)=3-x\:for\:2<x\le 3$ Determine $g\left(x\right)=fof=f\left(f\left(x\right)\right)$. Am a bit confused with the domain and range of ...
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1answer
29 views

Man-min inequality

It is known that $\underset{x}{\max} \underset{y}{\min} f(x,y) \leq \underset{y}{\min} \underset{x}{\max} f(x,y)$ . When does equality hold in this expression?
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0answers
15 views

Complex Plane - Analytic Function

I am trying to understand the definition of an analytic function and how to solve for it's domain. I understand that for $f(z) = {1\over z}$ the function is analytic on the complex plane except for 0. ...
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1answer
21 views

Need help with an algorithmic function [on hold]

Consider the following claim: for any positive constant c, f(cn) ∈ Θ(f(n))? Either show the claim is true or give a counterexample.
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2answers
30 views

Need help ordering a list of functions

List the functions below from lowest order to highest order. If any two or more are of the same order, indicate which. $n$, $n^3$, $2^n$, $\ln n$, $n^2$, $\ln^2 n$, $\sqrt n$, $2^{n−1}$, $\ln n$, ...
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0answers
21 views

Continuous functions using interval notation

Let $f(x) = \sqrt{x-2}$. Use interval notation to indicate where $f(x)$ is continuous. I don't know how to solve it.
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1answer
26 views

f is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^−1 (V ))$ = V prove?

$f$ is surjective ⇐⇒ $∀V ⊂ Y$, $f(f^{−1} (V ))$ = $V$ This is an assertion and i said it was true. But i am confused as to what is referred to as the domain and range in this question. I would say ...
3
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1answer
53 views

Find all functions $f:\Bbb Q\rightarrow\Bbb Q$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y\in\Bbb Q$

Find all functions $f:\Bbb Q\rightarrow\Bbb Q$ satisfying $f(x+y)+f(x-y)=2f(x)+2f(y)$ for all $x,y\in\Bbb Q$ I don't know how to proceed, any help would be really appreciated..
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1answer
22 views

If $f: B \rightarrow C$ and $g: A \rightarrow B$ be two functions let $h = f \circ g$. Then for $h$ to be onto what can we say about $f$ and $g$?

Let $f: B \rightarrow C$ and $g: A \rightarrow B$ be two functions and let $h = f \circ g$. Given that $h$ is an onto function, which one of the following is TRUE? (a) $f$ and $g$ should both be ...
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1answer
35 views

What is the series of the function 3 / ( 1- x^4)

I know that $f(x) = \frac{1}{ 1-x } = \sum_{n=1}^\infty x^n$. We can find that $g(x) = \frac{1}{ 1-x^4 } = \sum_{n=1}^\infty (x^4)^n = \sum_{n=1}^\infty x^{4n}$. Does the sum converge? what is the ...
2
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1answer
59 views

Is $f(x) = \infty$ a function?

Recently, while solving a problem where a certain set of functions $f:\mathbb Z^+ \rightarrow \mathbb Z^+$ had to be found given a number of conditions, I noticed that $f(n)=\lim_{a\to+\infty} a$, ...
2
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1answer
31 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
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2answers
41 views

Functions and limits two directions

A really cunning question which I can't seem to solve. If $\lim _{x\to \infty }\left(f'\left(x\right)\right)=0$ does it necessarily means that $\lim _{x\to \infty ...
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0answers
52 views

Question about Riemann zeta function + my proof

First let me say that I am 16 years old so I am not very professional in math. English is also a second language so I apologize for any mistakes. Now i have been reading about the Riemann zeta ...
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1answer
18 views

Converting Recursive Function into Closed/Explicit Form

so I have this recursive function here: $\forall n>1,f(n) = 2(f(n-1)) + n-1$, (where it is $0$ when $n$ is less than $1$) So I have tried to use iteration for this but it just gets more ...
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1answer
16 views

generalization of midpoint-convex

Let f : (a,b) → R is a midpoint-convex function (I didn't say continuity). Here I'd like to verify following inequality ""directly"". f( (x1+x2+x3)/3 ) ≤ (f(x1)+f(x2)+f(x3))/3 .. I can easily ...
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1answer
36 views

How to prove this? “For all sets A,B⊆D and functions f:D→R, we have f(A∩B)⊆(f(A)∩f(B)).” [duplicate]

Here's my attempt: f(A∩B) = f({x|x∈A∧x∈B}) = {f(x)|x∈{x|x∈A∧x∈B}} f(A)∩f(B) = f({x|x∈A}) ∩ f({x|x∈B}) = {f(x)|x∈{x|x∈A}} ∩ {f(x)|x∈{x|x∈B}} = {x|x∈{f(x)|x∈{x|x∈A}}∧x∈{f(x)|x∈{x|x∈B}}} And now I'm ...
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vote
2answers
28 views

Do we simplify the Proof by Contradiction?

Prove the following by contradiction: Suppose $a,b\in\mathbb{Z}$. If $4|\left(a^2+b^2\right)$, then $a$ and $b$ are not both odd (in other words, $a$ and $b$ are even) So, I did this: Assume $a$ ...
0
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3answers
33 views

Find real functions knowing their values for all natural point

Consider a function $f(x) : \mathbb{R} \rightarrow \mathbb{R}$ smooth enough such that $f(\mathbb{N}) \subseteq \mathbb{N}$. Is there some methodologies to find another function $g(x): \mathbb{R} ...
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4answers
41 views

Proving onto and 1-1 functions

I understand the 1-1 function side of things, but I still don't really get how to prove that the function is onto Question: Prove that the function $f:\mathbb{R}-\{2\} \to \mathbb{R}-\{5\}$ defined ...
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3answers
56 views

Does f(x) = g(u)?

If $f(x) = x + \sqrt{2-x}$ and $g(u) = u + \sqrt{2-u}$ is it true that $f = g$? I squared both sides $\sqrt{x + \sqrt {2-x}} = \sqrt{u + \sqrt {2-u}}$ $\sqrt{x} + 2-x = \sqrt{u} + 2-u$ I then ...
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1answer
50 views

Graphs of a functions: $ e^{x^2} , e^{1/x} $

I don't understand how to plot similar functions without a calculator. 1. $\arctan {1\over x-2} $ 2. $e^{x^2} , e^{1/x}, e^{2x\over1-x^2} $
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2answers
47 views

How do I read this definition of injective in English?

This is a different but related question to one I asked earlier. I link to it here: "To show that f is injective" - I don't get this statement I am pretty new to "functions" having ...
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0answers
20 views

verifying function one to one and onto

f: z-> z x z such that f(n) = (2n, n+3). Verify one to one or onto? I tried y = n+3 = f(n) Condition for one to one f(x) = f(y) so , x+3 = y+3 therefore , x = y Another try , If I take n = 1 an ...
2
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0answers
30 views

subspace of the Vector Space of real valued functions

This is a problem from Hoffman and Kunze's Linear Algebra 2nd edition. I am trying to determine whether or not a particular subset of the set of all real valued functions is a subspace. I've done ...
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1answer
30 views

Function Equivalent to a Constant Paradox

Say I define $z(x,y) = x^2+y = \text{constant}$ Then $\left(\dfrac{\partial z}{\partial x}\right)_{y} = 2x$ However, $\left(\dfrac{\partial \text{ constant}}{\partial x}\right)_y = 0$ Shouldn't ...
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2answers
69 views

Show that $f:\mathbb{R}-\{2\}\to\mathbb{R}-\{5\}$ with $f(x)=\frac{5x-1}{x-2}$ is bijective

Can anyone please help to explain the question and what actually $f: \mathbb{R} - \{2\}$ means ?? I know that bijection means one to one function and onto both. Any idea to start up with this ...
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1answer
23 views

Find the max volume using polynomials with the sum of the height and perimeter less than 100cm

I have to find out which shape of packaging for a fragile object has the most volume to fit the object and styrofoam packing. The sum of the height and the perimeter must be less than 100cm. There is ...
0
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1answer
19 views

Finding the vertical shift of a sinusoidal function

I'm currently studying sinusoids, I've been given a graph with a few key points and have been told to find a cosine function which fits it. When it comes to finding the vertical shift of the graph the ...
4
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1answer
33 views

Is there exist an additive but unbounded function?

I just learned that the function that is additive and bounded near $0$ on Real has the only form of $f(x)=cx$, where $c$ is a constant number. We say that a function $f$ is additive iff ...
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0answers
28 views

Functions commute with a given polynomial

Given a polynomial $f(x)\in \mathbb{C}[x]$,how to find(describe) functions(smooth or continuous or polynomial) that are commute(under composition) with $f(x)$? There are trvial ones :$x,f,f\circ ...
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2answers
62 views

Clarification of Identification [on hold]

This is more of an observation question. When you see $x$, In $f(x) = x^2$ And when you see $g(x) = x^3$ You automatically identify $x = x$ Wouldn't the $x$'s be off by a little bit? But ...
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0answers
17 views

Notation Question with regard to functions

Let $f : N → N$ Let $E(f)$ be the function defined by $E(f)(n) = 2^{f(n)}$. Does $E(f)(n)$ mean $E(f(n))$? or $E(f)(n)$?
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2answers
27 views

Simplify a function with a square root as the numerator

How would I go about simplifying this: $$\frac{\sqrt{x^4 + 3x^2}}{x}$$ thanks!
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1answer
35 views

Is there a way to re-write $\min(a,b)$ in terms of an analytical function?

Is there a way to re-write $\min(a,b)$ in terms of an analytical function? Also, if not, is there a nice analytic function that is a tight upper bound? This question is related to this question.
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2answers
25 views

Big O notation - proof

Is it true that $O(k(n) + m(n))$ is equal to $O(\max\{k(n), m(n)\})$? In one of papers on computational complexity I've found the following statement: $$O(\log(n) + n(\log S + \log V )) = O(n(\log ...
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1answer
25 views

a problem on functional equations [duplicate]

if fuction is defined from $N to N$ then can we say that it is not continuous as it is not defined for all $x$??a simple statement ,but this is stopping me from giving the solution to a question..pls ...
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0answers
14 views

Rounding up approximations when using iterative methods

When using iterative methods I have read that as soon as you get two successive approximations that round to the same number of decimal places, then all further approximations will round to that ...
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0answers
25 views

Bijectivity of a function $f(i,j)=\frac{(i+j)(i+j+1)}{2}+j$

Define $f\colon \mathbb N\times \mathbb N \rightarrow \mathbb N$ by $f(i,j)=\frac{(i+j)(i+j+1)}{2}+j$. how can I prove that f is bijective help please! I should prove that f is surjective and ...
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1answer
42 views

Limit of a function containing square root.

Q. $\lim _{x\to 0}\frac{\left(\sqrt{1-cos2x}\right)}{x}$ We can write this function as $\lim _{x\to 0}\frac{\left(\sqrt{2sin^2x}\right)}{x}$. Algebraically we have ...
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0answers
12 views

How to find a function from a matrix?

Suppose I have a matrix like this: H2 N2 G2 H1 0 3 8 N1 2 4 7 G1 1 5 6 How would I find a nice function $f(x, y)$ that ...
0
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0answers
12 views

Algorithm that adds three numbers in an array that performs in O(n^2) time

Note that this question extends on this previous question. Given an array A, and a value called value. Does there exist three ...
0
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1answer
26 views

How to combine two functions into one continuous function so it can be integrated/differentiated?

I have a function like this : $$f(x)=\begin{cases}x \in)-\infty, 1)&,\;\;f(x)=x^2\\{}\\x\in(1,+\infty(&,\;\;f(x)=x^3\end{cases}\;\;\;\;\;\;$$ as you can see, the function as a whole is ...
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2answers
14 views

Designing a fast algorithm which adds three numbers in array

Given an array A, and a value called value. Does there exist three elements in A where their ...
2
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3answers
68 views

Use of $\mapsto$ and $\to$

I'm confused as to when one uses $\mapsto$ and when one uses $\to$. From what I understand, we use $\to$ when dealing with sets and $\mapsto$ when dealing with elements but I'm not entirely sure. ...