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

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4
votes
6answers
65 views

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$

$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically? Using integration by parts I got the form: ...
1
vote
0answers
21 views

fn converges to f pointwise where all functions fn are bounded and f is unbounded: Is this example correct?

I am looking to find a sequence of functions $f_n$ that converges to a function $f$ pointwise, where all functions $f_n$ are bounded, but $f$ is unbounded. I have thought of an example where the ...
1
vote
1answer
9 views

What other types of distributivity are there?

When I say ‘Distributivity,’ I mean the way a number $x$ can be ‘Put in to’ some other function or the like. For example, to distribute $x$ into $\text{id}_y$, you simply have ...
4
votes
1answer
41 views

Solving the functional equation $f(x^2+f(y))=(f(x))^2+y$

Find all $ f : R\rightarrow R $ such that $f(x^2+f(y))=(f(x))^2+y, \forall\text{ x,y}\in R$ Thanks in advance!
0
votes
1answer
10 views

Writing a function in a part that is linearly dependent on the dependent variable

Let's say I have the function $f = f(x)$, Under what conditions am I able to write this function as follows: $f = c + h(x)x$ where $c$ is a constant, and $h(x)$ is some function depending on $x$.
0
votes
1answer
36 views

Evaluating $\lim_{n\to\infty}\int_0^1x^nf(x)\,dx$. [duplicate]

Let $f$ be a continuous function on [0,1]. Evaluate $$\lim_{n\to \infty} \int_0^1 x^nf(x)dx$$ My approach : Consider $\int x^nf(x)dx = \frac{f(x)x^{n+1}}{n+1} - \frac{1}{n+1}\int x^{n+1}f(x)dx$ ...
1
vote
1answer
21 views

Proof of Hyperbolic Functions

Find the proof:  (a) Use the definitions cosh(x)= 1/2(ex +e^−x) , sinh(x)= 1/2(e^x − e^−x) to express sinh(x + y) and cosh(x + y) in terms of cosh(x), sinh(x), cosh(y) and sinh(y). (b) Using the ...
1
vote
2answers
41 views

Can a real continuous bounded function on $ \Bbb{R}^{2} $ be expressed as a finite sum of products of real continuous functions on $ \Bbb{R} $?

Can a real-valued continuous bounded function on $ \Bbb{R}^{2} $ always be expressed as a finite sum of products of real-valued continuous functions on $ \Bbb{R} $?
-1
votes
0answers
24 views

Squares and Square Roots [on hold]

what are the laws/principles/rules to determine when to square or take a square root of a variable? To say it another way, how do you determine when you need to square or take a square root of a ...
5
votes
2answers
170 views

Why is $\sinh$ often pronounced “shine”?

I talked to some guys from the UK and they told me that they would pronounce $\sinh$ as "shine". I am not a native english speaker so I don't know, but in my country we call this function "sintsh" ...
0
votes
1answer
14 views

Precision needed in definition of unboundedness

This is a quick question about what it means for a function to be unbounded. Does it mean that the function tends to + or - infinity, or does it just mean that it has no limit?
1
vote
1answer
18 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
1
vote
1answer
33 views

How to generate integer random numbers that equal to another random number?

I am running a simulation in Excel, and need to generate a group of integer random numbers summing up to another random integer, how can I possibly do it? For instance I have an integer random number ...
2
votes
2answers
34 views

Can we solve recurrence relations indexed by R?

Recurrence relations are equations with functions from $\mathbb N\rightarrow \mathbb N$. For example given $a:\mathbb N\rightarrow \mathbb N$ (write $a(n)$ as $a_n$) solving the recurrence relation ...
0
votes
1answer
35 views

e Function Parity

I have look at the plot of the function $$(1+\frac{1}{x})^x$$ Is it an odd or even function? it seems like one flip on the Y-axis and on flip on the X-axis but unlike odd it is a flip upward Aren't ...
1
vote
1answer
30 views

Some conditions on $\tilde f(x,y)=\begin{cases}\displaystyle g(x,y) & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$

The following function $$f(x,y)=\begin{cases}\displaystyle\frac{x^2 y^2}{x^2+y^2} & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$$ is differentiable in the origin and ...
-3
votes
1answer
23 views

Given that $f(x)=\frac{5}{x-8}$ and $g(x)=\frac{9}{x+9}$, find [on hold]

(a) $(f+g) (x)=$ (b) $(f–g) (x)=$ (c) $(fg) (x)=$ (d) $(f/g) (x)=$ I don't know how to calculate the result of (x) .
0
votes
1answer
58 views

If $f(U)=0$ then what is possible?

Let , $U=\left(0,\frac{1}{2}\right)\times \left(0,\frac{1}{2}\right)$ and $V=\left(-\frac{1}{2},0\right)\times \left(-\frac{1}{2},0\right)$ and $D$ be the open unit disk centered at origin of $\mathbb ...
0
votes
0answers
11 views

Shifting a series of functions whilst maintaining symmetry

I have a function y = a*Exp[-(x - b)^2/2*c^2] + d*Exp[-Abs[-e*x]] + f Which is symmetrical when the coefficient of b is equal to 0 however it loses symmetry as ...
0
votes
1answer
16 views

The Average Rate of Change of Function

Here is the question: The following chart shows the growth of a crowd at a rally over a 3 h period. Time (in hours): $ 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 $ Number of people: $0 , 176, 245, 388, 402, ...
-1
votes
0answers
19 views

Eliminate the arbitrary Function of PDE

I need to solve this problem; Eliminate the arbitrary function $f$ from the equation: $f(x^2+y^2+z^2,z^2-2xy)=0$ I try this solution $u= x^2+y^2+z^2, \quad $ $\quad v=z^2-2xy, \quad$ so ...
2
votes
0answers
23 views

Inversion of a pairing function

I was reading this question on this site and I saw that the following pairing function was mentioned (a modified version of Cantor function): $$\langle x, y\rangle = x * y + ...
0
votes
1answer
23 views

Can I have a critique of this set theory proof/Advice on a similar proof?

This is an exercise from Mendelson's Introduction to Topology. The first part is to prove, given a function $\ f:A \rightarrow B$, that $\ X \subset f^{-1}(f(X))$ for all $\ X \subset A$. Here's my ...
3
votes
4answers
59 views

Prove that $[a/b]+[2a/b]+[3a/b]+…+[(b-1)a/b]=(a-1)(b-1)/2$

If a and b are positive integers with no common factor how to show that $[a/b]+[2a/b]+[3a/b]+...+[(b-1)a/b]=(a-1)(b-1)/2)$,where [.] denotes the greatest integer function? Im not able to understand ...
1
vote
1answer
17 views

Showing that a given function is convex.

I am trying to show that the function $f(x,\vec{y})=\alpha\ln(1+\exp(x+\vec{y}\cdot \vec{z}))+(1-\alpha)\ln(1+\exp(-x-\vec{y}\cdot\vec{z}))$ is a convex function of $(x,\vec{y})$ (where ...
4
votes
2answers
106 views

How do I prove that $ f(n) = (n + 1)! - 1 $ is an injective function?

I have this problem: Consider the function $f : \mathbb{N} \rightarrow \mathbb{N}$ defined, for every $n \in \mathbb{N}$, by $$f(n) = (n+1)! - 1$$ Prove that $f$ is injective. How do I ...
1
vote
1answer
38 views

Expanding function into Maclaurin series

How to expand that function below into Maclaurin series? Is it even possible? $$ f(x)=x^2+\ln\left(\frac{2x-3}{5-3x}\right) $$ I know that expanding into Maclaurin series requires function class ...
1
vote
1answer
21 views

Quadratic function with positive integral coefficients problem

Here is the problem statement: Let $f(x)$ is a quadaratic expression with positive integral coefficients such that for every $\alpha, \beta\; \epsilon\; \Re$, $\beta>\alpha$, $\int_\alpha^\beta ...
5
votes
0answers
40 views

Show that $1^k+2^k+…+n^k$ is $O (n^{k+1})$.

Let $k$ be a positive integer. Show that $1^k+2^k+...+n^k$ is $O (n^{k+1})$. So according to the definition of big-$O$ notation we have: $$1^k+2^k...+n^k ≤ n(n^k) = n^{k+1}$$ whenever $n>1$ Is ...
1
vote
0answers
29 views
+50

Find Functions That Can Be Inverted from Their Sums

I have the following situation:$$ f_1(x_1) + f_1(x_2) + f_1(x_3) + \cdots + f_1(x_n) = c_1\\ f_2(x_1) + f_2(x_2) + f_2(x_3) + \cdots + f_2(x_n) = c_2\\ \vdots\\ f_n(x_1) + f_n(x_2) + f_n(x_3) + ...
1
vote
2answers
23 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
-1
votes
1answer
27 views

A Problem Distribution Function

If I have a probability density function like this $w(x) = 1 - |x| $if $|x| \leq 1$ or $ w(x)=0$ if $|x|\geq 1$, what's the value of the distribution function F(x)? I mean that I calculated ...
0
votes
1answer
24 views

Manually plotting some particular graphs

How to plot graphs like these manually: 1) $f(x)=\ln(1+x^2)$ 2) $f(x)=\frac8{2+x^2}$ 3) $f(x)=\frac{\sin x}{\sqrt{1+\tan^{2}x}}+\frac{\cos x}{\sqrt{1+\cot^{2}x}}$ I have no idea how to plot the ...
1
vote
2answers
28 views

Continuity of $\mu(t)=\inf\{x \in \mathcal C : \kappa(x)=t\}$.

Let $\Delta = \{ 0, 1\}^{\mathbb N}$ be a Cantor set. Define $\theta : \Delta \to [0,1]$ by the formula $$\theta(x_1,x_2,\dots) = \sum_{n=1}^\infty \frac{2x_n}{3^n}.$$ Denote $\mathcal C = ...
0
votes
1answer
15 views

Difference Between Lyapunov and Strong Lyapunov Function.

Good Day everyone. I was assigned to show that given an autonomous system of Differential Equations and a function $V$, I need to show that $V$ is Lyapunov function. To show that $V$ is Lyapunov. I ...
0
votes
2answers
25 views

Let $f:[0,\infty]\to R$ be differentiable on $(0, \infty)$, and $f'(x)\to b$ as $x \to \infty$. Show that $\lim_{x \to \infty}\frac{f(x)}{x}=b$

This is actually part (c) of the original question. Part (a) asks to prove for any $h>0$, we have $\lim_{x\to\infty}\frac{f(x+h)-f(x)}{h}=b$. Part (b) asks to prove if $f(x) \to a$ as $x\to\infty$, ...
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votes
2answers
40 views

No. of surjections [on hold]

Find the number of surjections from a $3$-element set to a $2$-element set. Find a formula for the number of surjections from $ℙ_{k+1}→ℙ_k.$ Find a formula for the number of surjections from ...
0
votes
0answers
14 views

Support of Distribution Function

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
0
votes
3answers
64 views

Show that $(x^3+2x)/(2x+1)$ is $O(x^2)$

Show that $(x^3+2x)/(2x+1)$ is $O(x^2)$ The definition says: We say that $f(x)$ is $O (g(x))$ if there are constants $C$ and $k$ such that $$\mid f(x) \mid \leq C \mid g(x) \mid$$ whenever $x > ...
-2
votes
2answers
47 views

Flourish in functions

Let $f$ be a continuous function defined on $[-2009,2009]$ such that $f(x)$ is irrational for each $x\in[-2009,2009]$ and $f(0)=2+3^{\frac{1}{2}}+5^{\frac{1}{2}}$. Find the value of $f(2009)$. I am ...
1
vote
0answers
13 views

r(tau) within of H´s level curves

Im currently working on a math project and have one question regarding level curves. I´ve been given a function $$H(\theta,\omega)=\frac{1}{2}\omega^2-C\theta-\cos{\theta}$$ And I´ve shown that if ...
1
vote
1answer
51 views

How to prove this problem about supermodularity function?

The problem is as follows, and I have solved the subproblem (a), but haven't solved (b) yet. And for (b) the method I think about is proof by contradiction, but I get stuck before I could solve this. ...
2
votes
1answer
21 views

Iterated limits of $\frac{x-y}{x^3-y}$

Why it the following limits look like this: $$\lim_{x\rightarrow -1} \frac{x-y}{x^3-y}=1$$ but suprisingly $$\lim_{y\rightarrow -1} \frac{x-y}{x^3-y}=\frac{1}{1-x+x^2}$$I thought that after ...
0
votes
1answer
26 views

Integration: Find length of curve using NINT

Here are the questions - For question 4, part (b) gives a unit circle. But I'm unable to proceed with parts (a) and (c), since the curve is double valued for -0.5 Also, for question 6, integration ...
0
votes
1answer
39 views

Inverse of a function which contains an integration.

We are given a function $$Q(x) = \int^{\infty}_xf(t)dt$$ How do we find find $Q^{-1}$? If such a question has already been asked please, comment the link. Thanks in advance.
1
vote
1answer
33 views

Function that determines angular velocity?

I see that someone posted the same problem a year ago, but the answer didn't quite give enough info. Here's the question: A movie crew is working on a scene that involves filming a car moving at a ...
0
votes
1answer
20 views

Optimal Value & Uniform Distribution

In a simple setting, $w$ is uniformly distributed on $[0,1]$, R is a function of $wd$. I want to find optimal d in this expression, $aR-(d^2-1)/2$. When I try to find out optimal $d$ than it is $0$. ...
0
votes
0answers
13 views

respective values of two functions are “closer than expected”

let f,g be functions from the same finite set into the reals, let d be the mean distance between f(x) and g(x) for x in S, and let D be the mean distance between f(x) and g(y) for x,y in S; then D-d ...
3
votes
2answers
29 views

For what real values does $\phi(x):=1+x+ \dots + x^{2m-1}$ take the value $0$? What can you say about the sign as $x$ varies?

For what real values does $\phi(x):=1+x+ \dots + x^{2m-1}$ take the value $0$? What can you say about the sign as $x$ varies? I need help adding rigor to my observation to create a formal proof. ...
2
votes
5answers
54 views

Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable. Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$ Consider $g(x):= ...