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

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0
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2answers
281 views

Method of Lagrange multipliers to find all critical points of a function

I am having difficulties in understanding the steps/method required to find the critical points of a function using the method of Lagrange multipliers. I have read through my text book and tried my ...
1
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2answers
30 views

Redistributing Money in Monopoly

Is there a class of functions that satisfies the following properties? $\lim_{x \to -\infty}f(x)=-k, \lim_{x \to \infty}f(x)=k$ $f(x)<f(y) \Longleftrightarrow x<y $ ...
1
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2answers
53 views

Properties of bijections

If a bijection exists between set A={a1, a2, ...} and set B={b1, b2, ...} such that a1 maps to b1 and a2 maps to b2, etc., does this mean if we find a relationship R between a1 and b1 (i.e. f(b1) is ...
1
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1answer
52 views

Limit of average of real function

I need some hints regarding this exercice. if $f : [0, \infty)\rightarrow \mathbb{R}$ is a measurable function s.t $\lim_{x\rightarrow \infty} f(x) = a$, prove : \begin{align} \lim_{x\rightarrow ...
3
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3answers
40 views

Roots of real polynomial

$f(x)$ is a real polynomial. Show that $z=a+bi$ and $\bar z=a-bi$ have the same algebric multiplicity. I know that if $z=a+bi$ is a root of $f$ then $\bar z=a-bi$ is too, but don't know how to use ...
1
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2answers
83 views

Why are all non-polynomial functions are basically exponents?

There's paucity of really "original" functions in Math. Aside from power functions/ polynomials, really the only other function widely used is exponential. For example, $\log$ is simply inverse of ...
0
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1answer
38 views

Definition of the total variation of a function $g:\mathbb{R}\to\mathbb{R}$

if the total variation of a a real function $f:[a,b]\to \mathbb{R}$ over $\textbf{P}=\{a=t_0<t_1<...<t_m=b\}$ is $$ V^{a}_{b}(f)=\sup_{\textbf{P}}V(f,\textbf{P}) $$ where $$ ...
0
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1answer
47 views

Can we approximate $f(x) = \chi_{(0,\infty)}(x)$ by smooth monotone functions?

Given $f(x) = \text{sign}^+(x) = \chi_{(0,\infty)}(x)$, is it possible to a sequence $f_n$ such that $f_n$ is smooth, $f_n' \geq 0$ and $$f_n \to f?$$ In some sense of convergence? Preferably ...
1
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0answers
47 views

What is real periodic function

I would like to know what is real periodic function. I understand what is periodic function, but I do not understand what is "real" periodic function.
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1answer
44 views

Help me how to find the limit. In this case something is strange.

A fn(x) is a sequence of function. fn:[0,1]→R defined by fn(x) = n(1-x) × x^n. And I want to know limit of this function as n→∞. In my opinion we can see this like this n(1-x) × x^n. So as n→∞, ...
1
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1answer
56 views

Is there an actual function that “escapes” from zero after an amount of derivatives?

Suppose we have a function $f$ such that for all $i=2,...k: f^{(i)}=0$ but for $i\ge k+1$ we have $f^{(i)}\neq 0$. Can there be such functions in theory ? and is there an actual function that ...
0
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1answer
54 views

Functions of functions

Is there such a thing as the study of the calculus of functions (I can think of no better term for it!) eg: if $f_0(x)=\sqrt{n}\log(n)$ then $f_1(x)=\sqrt{\sqrt{n}\log(n)}\log(\sqrt{n}\log(n))$ ...
0
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1answer
50 views

Prove that $xy\le\frac{x^p}{p}+\frac{y^q}{q}$

Given $x\ge0$, $y\ge0$, $p>0$, $q>0$ such that $\dfrac{1}{p}+\dfrac{1}{q}=1$: Prove that $xy\le\dfrac{x^p}{p}+\dfrac{y^q}{q}$ starting by $xy=1$
2
votes
1answer
37 views

Being g a continuous function show that

$$ (f_n)_{n\in\mathbb N}, \quad x\in \mathbb R $$ $$ f_n(x) = \begin{cases} n+n^2 x & \text{if } x\in\left[-\frac 1 n , 0 \right], \\ n - n^2 x & \text{if } x\in\left[0,\frac 1 n \right], \\ 0 ...
2
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1answer
49 views

Domain of a composite function

I was given the question: Find the domain of the function $f(x)=\ln(\ln(\ln x))$ I found the answer by inspection: $\qquad D(\ln x)=(0,\infty)$ $\therefore\quad D(\ln(\ln x))=(1,\infty)$ ...
1
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1answer
30 views

Please, help me with this definition of a function

Let $A$ be a set. $y \in f(A)$ iff $f(x) = y$ for some $x \in A$. Suppose $A = \{2, 3\}$ and $f(x) = x^2$. Then $f(A) = \{4, 9\}. f(-2) \in f(A)$, but $-2 \notin A$. Is it contradicting the ...
2
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1answer
42 views

Frequency of Words in Document

I'm trying to figure this out: Would someone care to explain how one would go about using this function? More specifically, I don't understand the interval part, how does one count the intervals? ...
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0answers
47 views

$f'(x)$ to $f(x)$ is it possible without knowing the value of $f'(x)$ or $f(x)$

I don't know much math and i got stuck at a problem: I'm not sure if it possible how to do it. I must use hermite interpolation for the following: 'Find the polynomial interpolating the function $f$ ...
1
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2answers
66 views

Finding if a function with cases is differntiable on a point

Is $g$ differentiable on $x=0$ ? $$g(x)=\begin{cases}\dfrac{e^x-1}{x}&,x\neq0 \\ 1 &,x=0 \end{cases} $$ The derivative for $x\neq0$: $g'(x)=\dfrac{e^x(x-1)+1}{x^2}$, by taking the ...
0
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2answers
43 views

Prove the existence of exactly two maxima for a positive $L^1$ function

I have a function $f:\mathbb R \rightarrow \mathbb R^+$ with the following properties it is $L^1$ and $C^2$ it has one single extremum (maximum) at $x=0$ it is symmetric: $f(x)=f(-x)$. it is ...
1
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0answers
26 views

Tips for finding the range of a function

I am studying probability, specifically joint probability distributions. When computing sums or quotients I end up with things like this (when working with uniform random variables for instance): ...
1
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2answers
52 views

Calculate the radius of convergence [closed]

Being $\sum _{n=0}^{\infty \:}a_n\cdot x^n$, $a_1=a_0=1$ $a_{n+1}=a_n + a_{n-1}$ show that the radius of convergence is $\dfrac{-1+\sqrt{5}}{2}$ Thanks!
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1answer
55 views

Proving $\forall x\in \mathbb R$ and $ \forall A>0: \exists B>0 \ s.t \ \forall y\in \mathbb R, \ |y-x|\le A \Rightarrow |f(x)-f(y)|\le B$

Let $f:\mathbb R\to \mathbb R$ be a continuous function. Prove that $\forall x\in \mathbb R$ and $ \forall A>0: \exists B>0 \ s.t \ \forall y\in \mathbb R, \ |y-x|\le A \Rightarrow ...
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3answers
135 views

Alternative to epsilon-delta definition

Back in Multivariable Calculus, I remember my professor, when explaining why he decided to skip a rigorous definition of limit (the reason was that those of us that would continue in math would go ...
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0answers
27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
1
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1answer
13 views

relation between $|o(f)-g|$ and $|f-g|$

This question is similar to the one asked some hours ago. I have given three functions $f,g,h$ where $h(n)=o(f(n))$ and I know that $|f-g|<d<1$. Now I'd like to find an Expression for $|h-g|$. ...
2
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3answers
75 views

prove that $f(x)=\sum _{n=0}^{\infty}\frac{\cos(nx)}{2^n}$ is continuous

I refered that each fn is continuous because its the fraction of a continuous function by a number and so $f(x)$ that is the sum of continuous functions is continuous. Is it right?
2
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0answers
71 views

A real-valued separately continuous function discontinuous everywhere

I need an example of a real-valued separately continuous function that is discontinuous at each point. This is either a well-known folklore fact or a burning problem in separate versus joint ...
1
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1answer
25 views

Proving equality of functions using their restrictions

I have been going through Elementary Set Theory by Enderton and once again I am stuck on an exercise, which goes like this (p.88, exercise 27): Assume that $A$ is a set, $G$ is a function, and ...
1
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0answers
41 views

If $f$ is a continuous function $f:[0,1]\to[0,1]$ then there exists $x_0\in [0,1]$ such that $f(x_0)=2\sin x_0$

Prove/disprove: if a continuous function $f:[0,1]\to[0,1]$ then there exists $x_0\in [0,1]$ such that $f(x_0)=2\sin x_0$. Define: $g(x)=f(x)-2\sin x$ $g(\frac {\pi} 4)=f(\frac {\pi} 4)-2\sin\frac ...
0
votes
1answer
22 views

$O(f)-g = O(f-g)$: asymptotics of difference of functions

I have given three functions $f$, $g$, $h$ where it might be relevant that all these functions are bounded from above by $1$. I know that $$|f-g|=d$$ where $d$ may depend on $n$ and I know that ...
1
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3answers
25 views

Clarification of Functions

Let $f: \mathbb{Z}^2 \to \mathbb{Z}^2$ be defined as $f(m, n) = (m + n, 2m − 5n)$ . Is $f$ a bijection, i.e., one-to-one and onto? Since my function is mapped on the domain consisting of all integers ...
3
votes
1answer
79 views

1-to-1 correspondence between twin primes and $n^2-1$

I am trying to establish the one-to-one correspondence of twin primes to integers $n$ where $n^2-1$ has 4 divisors. It is clear to me that this is the case, since $$n^2-1=(n+1)(n-1)$$ where the RHS ...
5
votes
1answer
105 views

How find this sum of all distinct values of $f(2014)$

For all functions $f:\mathbb{R}\backslash\{0\}\to\mathbb{R}$, that satisfy $$f\left(x+\frac1x\right)f\left(x^3+\frac1{x^3}\right) - f\left(x^2+\frac1{x^2}\right)^2 = ...
0
votes
3answers
29 views

Confused about function terminology

Let $f: X \to Y$, $A \subseteq X$. So, $f(A) \subseteq f(X) \subseteq Y$. Then $f(A)$ is the image of the set $A$ and $f(X)$ is the image/range of the function $f$. Is the above correct? ...
0
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0answers
18 views

Bijective Functions between Multiple Dimensions [duplicate]

Do bijective functions exist that map from a function of one dimension to a function of another dimension? For example, does there exist a function $f : \mathbb{R^2} \rightarrow \mathbb{R^3}$ that is ...
3
votes
1answer
4k views

How can I read logarithmic scale?

I've got this histograms: How can I read that logarithmic scale? For example, on the histogram 1 there is approximately $10^{-3}$ value at y-axis at 2 value at x-axis. Does it meant that there is a ...
0
votes
4answers
94 views

Slightly equal functions

Can there exist two elementary functions $f(x)$ and $g(x)$ defined everywhere on the real axis such that, \begin{align} f(x)&=g(x)\qquad \text{if} \quad a\le x\le b\\ f(x)&\neq g(x)\qquad ...
0
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1answer
19 views

Calculating production based on growing population

I'm trying to find a solution to a programing problem, but the basis is Math. imagine a game where population grows continuously using a predetermined growth rate. Population in time t, denoted as ...
0
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2answers
84 views

Combination of continuous and discontinuous functions

I know that combining two continuous functions gives a continuous function, i.e., if $f(x)$ and $g(x)$ are continuous, then $f(x)\pm g(x)$, $f(x)\times g(x)$ and $f(x)\div g(x)$ are continuous ...
1
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1answer
72 views

Reciprocal Rule Question

$f(z) = \frac{4 z + 4}{z^2 + z + 1}$ I have a question about the reciprocal rule concerning derivatives. I want to work around doing the quotient rule for the above function if possible, but I am not ...
2
votes
2answers
627 views

Why can a discontinuous function not be differentiable?

I don't really understand why a discontinuous function cannot be differentiable. In Stewart's Calculus, the definition of a function $f$ being differentiable at $a$ is that $f'(a)$ exists. Earlier it ...
3
votes
1answer
57 views

Behavior of Two Functions

If two functions can be shown to agree at an infinite number of points, what additional information would be required to show that these two functions are equivalent? For example, if two polynomials ...
0
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1answer
30 views

Random points representation

Let's start with the following disclaimer: Math Noob Asking the question. I have a set of 30 (to be assumed) random numbers (they're actually not random at all): ...
2
votes
3answers
88 views

A certain polynomial P(x) , $x\in R$ when divided by $x-a, x-b,x-c$ leaves the remainders a,b,c respectively…

A certain polynomial P(x) , $x\in R$ when divided by $x-a, x-b,x-c$ leaves the remainders a,b,c respectively. Find the remainder when P(x) is divided by $(x-a)(x-b)(x-c)$ is (a,b,c are distinct) My ...
1
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1answer
67 views

$f(n)$ and $f(2^n)$ are co prime for all natural numbers $n$. Find all such polynomials.

Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^n)$ are co prime for all natural numbers $n$.
1
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1answer
56 views

If $f$ is an odd function, show that $f(0)=f''(0)=f''''(0)=…=0$

If $f$ is an odd function, show that $f(0)=f''(0)=f''''(0)=....=0$ and if $f$ is an even function, show that $f'(0)=f'''(0)=f'''''(0)=...0$ I do know why $f''(0)$ of an odd function is zero and why ...
17
votes
3answers
426 views

The function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$.

Show that if $f\in \mathcal{C}^3$ and $2\cdot\pi$ periodic then the function $f'+f'''$ has at least $3$ zeros on $[0,2\pi]$. My attempt : f is $2\pi$ periodic and $\mathcal{C}^3$, we have : ...
2
votes
3answers
68 views

Is fractional inverse of a function a known thing?

I know there's fractional Fourier transform, fractional derivative, maybe some other transformations generalized from being discrete to continuous. Now I wonder if there's any way to generalize a ...
3
votes
3answers
64 views

Finding the explicit notation of $f(n)$, based on it's recursive description.

I came across this problem on a HackerRank challenge. The function $f(n)$ is $1$ if $n = 0$ $2f(n - 1)$, if $n$ is odd $f(n -1) + 1$, if $n$ is even I solved the problem using a recursive ...