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

learn more… | top users | synonyms (1)

1
vote
2answers
28 views

If a function is convergent and periodic, then it is the constant function.

I have to prove that if a function f is convergent : $$ \lim\limits_{x\to +\infty} f(x) \in \mathbb{R}$$ and f is a periodic function : $$\exists T \in \mathbb{R}^{*+}, \forall x \in \mathbb{R}, f(x + ...
-4
votes
1answer
39 views

True or False? If true provide a proof. If false provide a counterexample. [closed]

A) Every subset of R has a least upper bound. B) If a sequence is not monotonic then it diverges. C) Let f : A -> B and g : B -> C be functions. g o f is surjective if and only if f and g are both ...
2
votes
1answer
23 views

The link between the monotony of a function, and its limit

Let's assume I have a convergent function f, as x approaches to $$+\infty$$. Is-it true to say that it exists a real x0, such that forall x>x0, f is either increasing or constant or decreasing ? (And ...
3
votes
2answers
18 views

“Greatest lower bound function”

If $f $ is a function continuous at $c, h $ is positive and $m$ is a function defined as $ m(h)=\inf \{ f(x): x \in [c,c+h] \}$ , how can I prove that the limit of $ m $ as $ h $ approaches $ 0 $ ...
2
votes
1answer
21 views

How do I denote a set of function values

I'm trying to denote the set of values given by the function $h(v, k)$ for all $v \in \{0, 1\}$ and $k \in \{0, 1, ..., 2^K-1\}$. I was thinking something like this: $$H = \{ h(v, k) : v \in \{0, ...
0
votes
1answer
68 views

Approximating a function by a sum of functions

I have a function $$ f(A,B,C):=\frac{\gamma-\delta \exp(\beta A+\omega B+C)}{1+\exp(\beta A+\omega B+C)} $$ where $\gamma, \delta, \beta, \omega$ are parameters. Do you know if there is a way to ...
3
votes
2answers
55 views

Show that $\mathbb{R}^{\mathbb{R}} = U_{e} \oplus U_{o}$ [duplicate]

A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called even if $f(-x) = f(x)$ $ \forall x \in \mathbb{R} $ A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is called odd if $f(-x) = -f(x)$ ...
1
vote
2answers
28 views

Intersection between a closed set and $y=x$ on $[0,1]$

I was given a little variant of "show that a surjective continueous function from $[0,1]$ to $[0,1]$ intersects with $y=x$ at least once" Let $P$ be a closed set of $[0,1]^2$ such that $\forall ...
3
votes
1answer
16 views

Convergence of functions with different domain

Question: Is there a concept of convergence for functions $f_n: D_n \rightarrow X$ with different domains to a function $f: D \rightarrow X$? I know concepts like uniform convergence or almost ...
0
votes
3answers
32 views

The only solution of the equation ${72_8!}/{18_2!}=4^x$ is $x=9$

Problem and Definitions If $n_a!:=n(n-a)(n-2a)(n-3a)\ldots(n-ka):n>ka$, how should I go about solving this?: $$\dfrac{72_8!}{18_2!}=4^x$$ Attempt ...
0
votes
2answers
24 views

Even and odd functions

Given $f(x)= \sqrt{1-\cos x}$. Period $0<x<2 \pi$ Is it a even function or a odd function? Whether the $f(x)$ has to be converted to square root of $2$ multiplied by $\sin(x/2)$.
4
votes
2answers
73 views

Why do we focus so much in math on functions (as a subclass of relations)?

Why is it that math so focuses on the subclass of relations known as functions? I.e. why is it so useful for us in nearly all branches of mathematics to focus on relations which are left-total and ...
0
votes
1answer
38 views

Prove or disprove: if $f$ is one-to-one and $g \circ f=h \circ f$ then $g=h$

I tried using a counterexample but couldn't find one. I also tried proving it by saying that $f(x)=f(y)$ since its one-to-one. Then I set up $g(f(x))=h(f(x))$ but didn't know what else I could do with ...
0
votes
1answer
31 views

Problem involving distances on a parametrically defined curve

SO the problem is stated below: A curve is defined paramterically by: $$x(t)=a\cosh t$$ and $$y(t)=b\sinh t$$ where a and b are positive constants and $-\infty < t < \infty$ The expression ...
0
votes
3answers
46 views

How do you reverse $\frac{100n(n+1)}{2}=c$ to find n given c?

I'm developing a game where the character experience needed by level is given by Gauss' formula multiplied by 100: $ \dfrac{100\mathrm{level}(\mathrm {level}+1)}{2}$. So the experience table is ...
0
votes
1answer
43 views

Prove or disprove: $f^{-1}(f(f^{-1}(Y))) = f^{-1}(Y)$.

Let $f: A \to B$ be a function, and $Y \subseteq B$. Prove or disprove: $f^{-1}(f(f^{-1}(Y))) = f^{-1}(Y)$. My textbook has a theorem that says: Suppose $f: A \to B$. Let $X \subseteq A$ and $Y ...
1
vote
2answers
65 views

Transforming the exponential function of a sum into the sum of functions

Is there a way to transform the function $$\exp(A+B+C),$$ where $\exp(\cdot)$ is the exponential function, into a sum $$f(A)+f(B)+f(C)?$$
2
votes
1answer
67 views

What are the double union ($\Cup$) and double intersection ($\Cap$) Operators?

Finale of THIS. Unicode says that $\Cup$ and $\Cap$ are double union and intersection, respectively. I was wondering if there was an actual operation that went with these symbols. If not, would these ...
0
votes
2answers
598 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
0
votes
1answer
13 views

Mean of a discrete function

I have a set of finite variables (~20) which when plotted are forming a decaying exponential function. I want to know is there a better way to calculate the average of this function than by just ...
0
votes
0answers
24 views

Equation for flat circular/lenticular surface

We know that $x^2+y^2= r^2$ is equation of circle curve but I want to draw a flat circular surface....not curve. i.e. to explain the problem-lets draw a flat circle not move it out of screen (positive ...
0
votes
1answer
349 views

Cancellation laws for function composition

Okay I was asked to make a conjecture about cancellation laws for function composition. I figured it would go something like "For all sets $A$ and functions $g: A \rightarrow B$ and $h: A \rightarrow ...
1
vote
0answers
25 views

Riemann-integrablity of monotonically increasing function [closed]

What is the proof of that every monotonically increasing function is a Riemann-integrable function?
4
votes
2answers
101 views

Is there a function $f\colon\mathbb{R}\to\mathbb{R}$ such that every non-empty open interval is mapped onto $\mathbb{R}$?

I wonder whether there is a function $f\colon\Bbb R\to\Bbb R$ with the folowing characteristic? for every two real numbers $\alpha,\beta,\alpha\lt\beta$, $$\{f(x):x\in(\alpha,\beta)\}=\Bbb R$$ ...
0
votes
1answer
38 views

Consider the function $\theta:\mathcal{P}(\mathbb{Z}) \to \mathcal{P}(\mathbb{Z})$ defined as $\theta(X) = \bar{X}$

Consider the function $\theta:\mathcal{P}(\mathbb{Z}) \to \mathcal{P}(\mathbb{Z})$ defined as $\theta(X) = \bar{X}$. Is $\theta$ injective? Is it surjective? Bijective? Explain. I know how to prove ...
0
votes
1answer
27 views

Properties of concave functions

I would appreciate any help or hints in proving this. I believe it derives from the properties of concave functions, but I don't know much about this. 1) Let $C \geq 0$ be a constant and $0 \leq x ...
1
vote
1answer
26 views

Multivariate function maximum criterion

Be a concave mutivariate function $f(\textbf{x})=\textbf{y}$. I observed the following conjecture: the maximum value of $f$ is achievable when all entries of $\textbf{x}$ are equal. How to prove such ...
0
votes
0answers
21 views

Besov norm in $W^{1,2}(\mathbb{R}^n)$

A well known result on Besov spaces is that $\Lambda_1^{2,2}(\mathbb{R}^n)=W^{1,2}(\mathbb{R}^n)$. One way to define this Besov space (without Fourier transform) is to consider $$ ...
0
votes
1answer
20 views

Inverse of shannon entropy

The shannon entropy of a bit $(p,1-p)$ is $$H(p)=-p\log(p)-(1-p)\log(1-p)$$. This is a well behaved function that uniquley assigns each state (up to permutation of its elements, i.e. ...
2
votes
3answers
47 views

A question about surjective functions.

I am looking for a sample of surjective function $f:X \to Y$ and a set $A \subseteq X$ such that $f^{-1}(f(A))\neq A$. Is the sample $f(x)=x^2, f^{-1}(x)=\sqrt{x}, X=\mathbb{R}, Y=[0, +\infty), ...
5
votes
2answers
64 views

Integrate a periodic absolute value function [duplicate]

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
2
votes
1answer
19 views

Recursive functions.

If you have a recursive function $$g(x) = f(f(x))$$ and you know that $$f(0) = 0, f'(0) = 1, f''(0) = 2$$ Will then $$g(0) = 0, g'(0) = 1, g''(0) = 2$$ ?
0
votes
0answers
16 views

Can there be a Dirichlet series that gives the functional inverse of the Riemann zeta function?

Can there be a Dirichlet series that gives the functional inverse of the Riemann zeta function? I will delete this question if it gets downvoted.
0
votes
0answers
23 views

can someone help me with completing the whole integration [duplicate]

can someone please help me with this calculus problem. My lecture used sin2(pi/3) when he plotted the numbers using the 0 to 3 period and the wave landed on 0.5. i cant figure out what number i ...
1
vote
0answers
11 views

Rearrange seasonalized von Bertalanffy growth equation to solve for age

I'm hoping someone can help me rearrange the following equation to solve for $t$: $L_t = L_{inf} * (1-e^{-K*(t-t_0) + (CK/2\pi*sin(2\pi*(t-t_s))) - (CK/2\pi*sin(2\pi*(t_0-t_s))) })$ The ...
0
votes
1answer
34 views

Express recurrence in closed form

I am having trouble understanding the process of expressing the following recurrence in its' closed form. First of all, I do not really understand what "closed form" means. If someone could elaborate ...
0
votes
1answer
83 views

Questions about $f(n)=3+\frac{12}n$

Experimental Psychology: To study the rate at which animals learn, a psychology student performed an experiment in which a rat was sent repeatedly through a laboratory maze. Suppose the time in ...
1
vote
1answer
12 views

How to prove the function is smooth after define it in a removable singularity

Consider the function $f(z)=\frac{1}{z} - \frac{1}{sinz}$ on $[0,1]$. If we define it is 0 at 0 as its limit. How to show the function after adding definition on $0$ is smooth? It may be apparent, ...
0
votes
1answer
22 views

Show that a non-constant entire function has a dense image.

Let $f$ be a nonconstant entire function and $U$ be an open set in the plane. Show that there is a $z_0$ such that $f\left(z_0\right)\in U$. This question is an exercise for the Maximum Modulus ...
0
votes
2answers
45 views

Finding an interval $I \subset \mathbb{R}^+$ such that $\phi$ is decreasing on $I$

Given $0<\alpha<\beta<1$, we define a function $$ \phi(x) = x - x \left[\frac{x^\alpha + x^\beta+1}{\alpha(x^\beta+1)+\beta(x^\alpha+1)} \right], $$ I am trying to find additional sufficient ...
0
votes
1answer
30 views

Literally draw a function and convert it to x and y points arrays

I'm looking for free tool (eventually commercial) for drawing plot - I mean literally: with pencil. This plots are going to be for simple demonstration. This tool should convert my drawings to x and y ...
1
vote
3answers
23 views

disproving smallest integer function limit

prove that the limit of this function does not exist : $\lim_{x \to 1} \lfloor x \rfloor$ I know that the value of $\lfloor x \rfloor$ when $ x \to 1^-$ is $0$, the value if $\lfloor x \rfloor$ ...
0
votes
2answers
12 views

Study of a parametric function

I would like to study this function for $x\geq 0$, $\forall b,d \in \mathbb{R}$: $$ y=\frac{b+dx}{1-b-dx} $$ Can I say that it is monotone increasing (decreasing) over $x$ in its domain for $d>0$ ...
2
votes
2answers
146 views

Squeeze theorem with infinite limits

Let f,g be functions that are defined in the area of $x_0$ (Except $x_0$ itself) $f(x) \ge g(x)$ Given the limit $ \lim_{x \to x_0}g(x) = \infty $ Prove that $ \lim_{x \to x_0}f(x) = \infty $ It ...
1
vote
1answer
17 views

What is the chance that a PDF with compact support is concave?

Relevant questions and answers, in chronological order: When do equations represent the same curve? Find a smooth function with prescribed moments Does a sequence of moments determine the function? ...
1
vote
0answers
33 views

Can we ever have E(argmin(f)) = argmin(E(f))?

Consider a parametric real-valued function $f_{\boldsymbol{\alpha}}:\ \mathbb D^N \rightarrow\mathbb R$ whose parameters $\boldsymbol\alpha$ vary according to some distribution $\psi$, and $\mathbb D$ ...
1
vote
1answer
18 views

Proof by induction for a recursive function f

Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive case: $\operatorname{f}(x) = ...
0
votes
1answer
373 views

Prove tha $\sin^26x-\sin^24x=\sin2x\sin10x$

$$\sin^26x-\sin^24x=\sin2x\sin10x$$ Please help me to solve the above trigonometric function as I am trying to solve this question since last an hour.
4
votes
2answers
351 views

How to prove a function is not onto?

Let $f : Z\to Z$ be the function defined by $f(x) = 3x + 1$. Prove that $f $ is not onto, using a proof by contradiction. (Choose an integer $n$, and then prove ($\forall m \in Z$)($f(m) ≠ n$) by ...
2
votes
1answer
44 views

Applying Newton-Raphson method to $a\cdot b^{-2}=c\cdot d^4+e\cdot f(d)$

I am familiar with the method and it's application in classic problems, but I have troubles tackling the function I need to solve with it. So, variables in problem: Real numbers, all are known ...