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

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1answer
45 views

Degree of Equations [closed]

A) Which variables in the formula $V = \pi r^2 h$ would you need to set as a constant in order to generate: a linear equation? a quadratic equation? B) How should $r$ and $h$ be ...
0
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1answer
40 views

Guess a function that fits empirical data

This is my empirical data: Which function it looks like? I tried to guess (1) a dumped (exponential decaying) sinusoidal, but it does not oscillate after overshoot; (2) a sigmoid, but it oscillate ...
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3answers
40 views

find extrema of $2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$

$$f(x,y,z)=2-\left(z-\sqrt{x^2+y^2}\right)^2+\left(z-\sqrt{x^2+y^2}\right)^3$$ Find maximum and minimum of the function.
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0answers
45 views

implicitly define a function

The first part i made $u=\frac{z}{x}$ and $v=\frac{y}{x}$ and after calculating the partial derivatives $\frac{dz}{dx}$ and $\frac{dz}{dy}$ The second i have no idea how to do it
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1answer
27 views

Slope of a function very much less than the function

I was working on a Cosmology problem and got stuck at this approximation used in a paper. Fundamentally the approximation is, $\frac{df(x)}{dx} \ll f(x)$. Now I can't understand how to imagine this ...
2
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3answers
48 views

Find the coefficients $a, b, c$ and $d$ so that the curve shown in the accompanying figure is the graph of the equation.

Find the coefficients $a, b, c$ and $d$ so that the curve shown in the accompanying figure is the graph of the equation $y = ax^3 + bx^2 + cx + d$. I have no clue how to solve this. This looks like ...
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1answer
21 views

Function with similar properties

Suppose I have a function $f$ and derive another function from it with similar properties. For example I have that my new function is zero when the other function is zero. I would still like to use ...
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1answer
55 views

$f(tx,ty,tz)=t^n \cdot f(x,y,z)$ [closed]

For $t>0$ and $(x,y,z)\in \mathbb{R}^3$ show that if $f(tx,ty,tz)=t^n \cdot f(x,y,z)$ then $$x*\frac{\partial }{\partial x}f+y*\frac{\partial }{\partial y}f+z*\frac{\partial }{\partial z}f=n ...
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2answers
32 views

Why is this inequation correct?

$$\sup |f(x)| -\inf |f(x)| \ge \sup f(x) -\inf f(x).$$ How can you show that it is indeed true?Sup is the lowest upper bound and inf is the greatest. lower bound
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2answers
49 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 ...
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2answers
22 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 $ ...
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2answers
46 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
37 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
36 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 ...
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2answers
77 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 ...
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1answer
22 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 $$ ...
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1answer
33 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 ...
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0answers
41 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
31 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→∞, ...
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1answer
54 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 ...
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1answer
44 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 $f_1(x)=\sqrt{\sqrt{n}\log(n)}\log(\sqrt{n}\log(n))$ ...
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1answer
47 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 $xy=1$
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1answer
35 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
20 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: $\qquad D(\ln x)=(0,\infty)$ $\therefore\quad D(\ln(\ln x))=(1,\infty)$ ...
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1answer
28 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 ...
1
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1answer
33 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
43 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 $f$ ...
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2answers
52 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 ...
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2answers
34 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 ...
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0answers
18 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): ...
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2answers
44 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
39 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
78 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 ...
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1answer
12 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 $|h-g|$. ...
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3answers
64 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?
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0answers
30 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 ...
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1answer
16 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 ...
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0answers
27 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 ...
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1answer
19 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 ...
0
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1answer
60 views

Linearly approximating a curve [closed]

I'm trying to approximate y = x^3 with straight lines. One of the points at the end of two of the linear segments is (2,8). The linear segments are 22.5 units long. What are the coordinates of the ...
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3answers
24 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
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1answer
67 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
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1answer
79 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 = ...
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3answers
26 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
15 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 ...
1
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1answer
29 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
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4answers
75 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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0answers
24 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
0
votes
1answer
15 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 ...