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

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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 $ ...
3
votes
4answers
55 views

What is the difference between a surjective and a continuous function?

What is the difference between a surjective and a continuous function? If a function is surjective then it takes all values so it is continuous and also if a function is continuous then it takes all ...
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 $f(-x) = -f(x)$ ...
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
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 $\sin(x/2)$.
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 ...
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 ...
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 ...
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 ...
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
24 views

Finding the domain of a fourth-root of an equation with a term to the fourth

Hi – I've got this question about finding the domain of a function, and I got the answer, but the method I used is quite different from the explanation provided. My question is: Is my method flawed in ...
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 ...
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 ...
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 $$f(A)+f(B)+f(C)?$$
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 ...
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?
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 ...
0
votes
0answers
25 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
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
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 $$g(0) = 0, g'(0) = 1, g''(0) = 2$$ ?
2
votes
3answers
49 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), ...
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, ...
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 ...
1
vote
2answers
23 views

Number of monotonic set functions from all the subsets of some finite set to 0 or 1

Let $N=\{1,2,\ldots n\}$ be some finite set. Let $f:P(N)\rightarrow\{0,1\}$ be a function such that $A\subset B\rightarrow f(A)\leq f(B)$ I'm trying to find an upper bound to the number of such ...
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 ...
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
1answer
35 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 ...
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 ...
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
1answer
31 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 ...
0
votes
2answers
46 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 ...
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 ...
0
votes
4answers
55 views

Applying a function to both sides of an equation doesn't change it?

Why is it that applying a function to both sides of an equation doesn't change it? Can this be proven? Can you point to some material to read more about this?
0
votes
2answers
33 views

What does a mini circle between f and h(x) mean?

I am currently doing a math problem and have come across an unfamiliar notation. A mini circle between $f$ and $h(x)$ The question ask me to find for 'the functions $f(x)=2x-1$ and $h(x)=3x+2$' $$f ...
0
votes
4answers
61 views

How do I figure out if a 'function is odd or even'?

I am currently doing an highschool math problem and I do not know what the question is asking for when it asks 'state which functions are odd and which are even for the below'. $f(x)= x^2+1 $ ...
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
0answers
14 views

Specific function

I'm looking for a functions with 2 parameters (to plot in 3D) which will satisfy the following criteria: ...
1
vote
1answer
39 views

Limsup and liminf of a function

Let $k\in(0,1)$ be fixed and $L\in \mathbb{R}$ is finite. If $\limsup_{x\to\infty}f(kx)=L$ and $\liminf_{x\to\infty}f(\frac{x}{k})=L$ then is it possible to say $$\lim_{x\to\infty}f(x)=L.$$
0
votes
1answer
26 views

Proof by induction for a recursive function

Really having a tough time doing this question: Consider the function $\operatorname{f}: \Bbb N \to \Bbb N$ defined recursively as follows: 1) Base case: $\operatorname{f}(0) = 0$ 2) Recursive ...
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 ...
0
votes
2answers
35 views

Can functions have output values that aren't mapped?

I'm working with injective functions and I'm trying to prove that the cardinality of the domain of a function is equal to the codomain of the same function. Each input value is mapped to a different ...