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

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0
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3answers
77 views

$f(x)$ is a periodic function. What is its period?

Suppose that $f(x)$ is a periodic function. If we have $$\forall x :f(x+346)=\frac{1+f(x)}{1-f(x)}$$ what is its fundamental period ?
0
votes
3answers
34 views

How to find graph of the sum of two functions

Suppose I know the graphs of two functions $f(x)$ and $g(x)$. How can I find the graph of $h(x)=f(x)+g(x)$? What are the rules to be followed ? P.S. In case my question seems silly,at least provide ...
0
votes
2answers
49 views

A question about Idempotent functions [on hold]

some functions are such that $f\circ f(x)=f(x)$ like these 1) $$f(x)=x \implies f\circ f(x)=x=f(x)\\$$ 2)$$f(x)=\lvert x\rvert \implies f\circ f(x)=\lVert x\rVert=\lvert x\rvert=f(x)\\$$ 3) ...
6
votes
4answers
928 views

When I was teaching absolute function properties, I suddenly made this question …

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right ...
8
votes
3answers
741 views

Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
0
votes
1answer
37 views

General Question about number of functions

I am wondering if there is any sort of algorithm , or if not, at least some general approach to the following; Lets say we have two finite sets $$A=\{a_1,a_2,…a_n\}$$ and $$B=\{b_1,b_2,…,b_m\}$$ ...
0
votes
2answers
26 views

Show that an irrationally periodic function is also a constant function [duplicate]

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$, and any real number $x$ we have $f(x)=f(x+r)$. Show that $f$ is a constant function.
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votes
1answer
28 views

Characteristics of a logarithmic function [on hold]

For my math course I have been given a practice question which I am not sure how to go about. For the function: $$g(x) = 2 \log_{10} (x +1)$$ i) State the domain and the equation of the asymptote. ...
1
vote
3answers
82 views

What is meaning of this question and how to solve it?

I am stuck with understanding the meaning of the question, which states: Show that $\cos(n\theta)=f_n(\cos\theta)$ for polynomials $f_n(x)$ satisfying $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
1
vote
1answer
41 views

How to Prove It Exercise 7.2.5

Prove that ${}^{\mathbb{Z}^+} \mathcal{P}(\mathbb{Z}^+) \sim \mathcal{P}(\mathbb{Z}^+)$ where ${}^A B$ means the set of all functions $f:A \rightarrow B$ and $\mathcal{P}(A)$ is the power set of $A$. ...
2
votes
2answers
43 views

Function on half plane, continuity

let $\mu$ be a finite positive borel measure on $\mathbb{R}$ and let $\mathbb{H}$ denote the upper half plane $\{(x,y) \in \mathbb{R}^2: y > 0\}$. consider the functions ...
0
votes
3answers
40 views

Find bounded function satisfying f(0)=0, f'(0)=0, and bounded first and second derivatives

I am looking for a bounded funtion $f$ on $\mathbb{R}_+$ satisfying $f(0)=0$, $f'(0)=0$ and with bounded first and second derivatives. My intitial idea has been to consider trigonometric functions or ...
7
votes
2answers
109 views

Proving that the product of two numbers (in $\mathbb{R}$ or $\mathbb{C}$) is a continuous function.

This is what is given in the textbook, I will highlight what is confusing me: Product in field $\mathbb R$ or $\mathbb C$,on $X \times X$ defined as: $$(x,y)\mapsto xy$$ (Let indicate that map with ...
-3
votes
0answers
29 views

logarithmic function? [on hold]

Zeus Industries bought a computer for $2149. It is expected to depreciate at a rate of 25% per year. What will the value of the computer be in 3 years? Round to the nearest penny. Do not type the ...
-1
votes
1answer
34 views

prove $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$

Can I prove the following (with or without assumptions, e.g. all the elements in $\mathbf{a}$ or $\mathbf{b}$ are positive? $\max \mathbf{a}^T \mathbf{b}_i \leq \mathbf{a}^T \mathbf{c}_i$ where ...
0
votes
0answers
15 views

Is it possible to prove $\arg\min_a f(\max(\mathbf{a}^T\mathbf{b}_i)) = \arg\min_a f(\mathbf{a}^T\max(\mathbf{b}_i))$

I have the following optimization problem, $ \arg\min_\mathbf{a} f(\max(\mathbf{a}^T\mathbf{b}_i))\;\; i=1, \dots ,N$ where $\mathbf{a}$ and $\mathbf{b}_i$ are vectors of dimension $d$. Let $B = ...
0
votes
1answer
52 views

How to prove the following inequality? (or a counter example)

We know that we have $[\int |f(x)|^{p} \mu(dx)]^{1/p}\leq [\int |f(x)|^{q} \mu(dx)]^{1/q}$ when $p\leq q$, where $\mu$ is a probability measure and $f$ is a smooth function. Do we in general have the ...
2
votes
1answer
19 views

Determine the domain and range of the following relations using set builder notation.

I have been given the following relations to find the domain and range of using builder notation. I am just beginning to learn the whole concept of set builder notation, and I am running into a ...
2
votes
0answers
50 views

Is it possible to prove that a relation is a function or not by using derivative?

There several ways to prove if a mathematical formula is a function or not : First: To find 2 or more $X$'s that have the same $Y$ assigned to them . Second: To assume that we put $(x_1,y_1)$ and ...
1
vote
1answer
20 views

How to find a function that satisfies 2 conditions

I am solving a partial differential equation by separation of variables. Part of the solution requires finding a function that meets the following criteria. f(L,t)=C f(0,t)=A*cos(at+b) I was ...
1
vote
2answers
29 views

function such that the sum of previous f(x) is smaller than f(x)

Just out of curiosity: is there a function $f$, such that $ \forall x, \sum_{x'<x} f(x') < f(x) $ sum or integral...
1
vote
1answer
14 views

Comparison of arbitary conway chains (in particular a chain with $m$ $m's$) to $f_{\omega^2}(n)$

Wikipedia describes the Conway chained arrow notation and the fast growing hierarchy. I learnt that the function $f(n):=\large f_{\omega^2}(n)$ has the same growth rate as the function ...
2
votes
2answers
24 views

creative method to obtain range of newton function ?!

I am searching for more proof that the range of $y=\frac{x}{x^2+1}$ is $ \frac{-1}{2}\leq y \leq \frac{+1}{2}$ these are my tries : domain is $\mathbb{R}$ first : $$y=\frac{x}{x^2+1}\\yx^2+y=x ...
5
votes
5answers
119 views

$f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h $ find f(7)

Problem : $f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h$ Given that : $f(1)= 1, f(2) =2 , f(3) = 3, f(4) =4, f(5)=5, f(6) =6$ find $f(7) =?$ My approach: We can put the values of $f(1) = 1$ in the ...
0
votes
4answers
68 views

Is it always possible to converge from an integer to another integer? [on hold]

Let's say I'm given a fixed integer, I. I'd like to know if it is always possible to find a function, that starting from any random integer J will converge to or oscillate reasonably close (let's say ...
2
votes
2answers
27 views

Why is this proof for an arbitrary function constrained to a constant one?

Sorry if this seems trivial, I'm having some difficulty understanding a proof. I'm doing exercise 5.1.14 of Velleman's How to Prove It and a solution posted in this question, including the comments, ...
3
votes
3answers
41 views

Prove that equality holds only if $f$ is one-to-one.

I am just looking for a hint. Not a solution as I am just trying to solve these for fun. Let $f:A \rightarrow B$ with $A_0 \subset A$ and $B_0 \subset B$. Show that $$A_0 \subset f^{-1}(f(A_0))$$ ...
3
votes
1answer
48 views

What is the precise difference between functions and operators?

I have heard affirmatively that all operators are functions, but not all functions are operators. But at the same time I have heard that functions map numbers to numbers, whereas operators map ...
0
votes
2answers
45 views

Prove that the unique zeros of $f(x,y)=a x +(1-a)y+xy$ when $x,y\in[0,1]$, is $x=y=0$.

Prove that the unique zeros of the two-variables function: $$f(x,y)=a x +(1-a)y+xy$$ when $x,y\in[0,1]$, is $x=y=0$. Here, $a$ is a parameter between 0 and 1. I have no idea where to start. Any ...
0
votes
1answer
29 views

Can a limit of multivariable function can be taken componentwise?

Is there a theorem saying that $\lim_{(x,y)\rightarrow(p,q)} f(x,y)= \lim_{x\rightarrow p}(\lim_{y\rightarrow q} f(x,y))$? If so, could someone link me to a proof of it or give me a proof? Edit: So ...
19
votes
11answers
3k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
0
votes
1answer
75 views

Prove expectation finite

Let $ \{b_n\} $ be a sequence of non-zero complex numbers. We have $ N(t)=\#\{n \geq 1:|b_{n}|\leq t\} $, $ \displaystyle\limsup_{t\rightarrow\infty} N(t)/t^p<\infty (1\leq p <2)$, for $ X_1\in ...
2
votes
2answers
34 views

Range of an inverse trigonometric function

Find the range of $f(x)=\arccos\sqrt {x^2+3x+1}+\arccos\sqrt {x^2+3x}$ My attempt is:I first found domain, $x^2+3x\geq0$ $x\leq-3$ or $x\geq0$...........(1) $x^2+3x+1\geq0$ ...
1
vote
1answer
77 views

Is it possible to define $x+x+x+x…x$ times? [duplicate]

Is it possible to define $x+x+x+x...x $ times? I need to compute its derivative. It differs from the derivative of $x^2$. It evaluates to $x$ via sum of derivatives.
2
votes
1answer
40 views

Function of sin x

Give that $f(x)=\sin x$ for the domain $0\leq x \leq k$, find the greatest value of $k$ for which $f(x)$ has an inverse. Is the answer $\frac{\pi}{2}$?
0
votes
1answer
16 views

How can you test if a function is bounded by another?

I am trying to learn about complexity theory, which states that $f(n)$ is in $O(g)$ if, for some $C > 0$, $f(n) \leq C\cdot g(n)$ for all $n \in \mathbb{N}$. That's well and good; it makes sense to ...
0
votes
2answers
23 views

Find k for which the equation has equal roots.

I find to find the value of k in terms of $\alpha$ and $\beta$ . I rearranged the equation $x^2 +kx-1=2k+x$ into $x^2+(k-1)x-(1+2k)=0.$ I then found $\alpha$$\beta$ to equal$-1-2k$. and $\alpha + ...
0
votes
1answer
14 views

Naka-rushton function

I am trying to figure out how to transform the naka-rushton equation to a S-shaped function. The naka-rushton equations is defined by \begin{equation} R(C) = R_{max}\frac{C^n}{C^n+K^n}+b, ...
0
votes
2answers
37 views

inverse a function with exponential and first degree polynom

I need some help to inverse this function: $$ y = a(e^{bx}-1) + cx + d $$ with $y(0)=d$ and $y(k)=0$ where $k$ is a constant. I don't know how to proceed. Thanks.
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votes
2answers
25 views

Properties of a certain binary relation [on hold]

$R$ is a binary relation function $(x,y) \in \mathbb R^2$. If $$ R = \{(x, y)\in\mathbb R^2\mid \lfloor x\rfloor = \lceil y\rceil\} $$ then: Is $R$ reflexive, irreflexive or neither Is $R$ ...
2
votes
1answer
58 views

How to find the Summation S

Given function $f(x)=\frac{9x}{9x+3}$. Find S: $$ S=f\left(\frac{1}{2010}\right)+f\left(\frac{2}{2010}\right)+f\left(\frac{3}{2010}\right)+\ldots+f\left(\frac{2009}{2010}\right) $$
0
votes
0answers
57 views

A weird problem on expected value of a random variable [on hold]

Let $X$ be a discrete random variable taking values $x_1, x_2$, ... with probabilities $p_1, p_2$, ... respectively. Then the expected value of this random variable is $E(X)=\sum_{i=1}^{\infty }x_i ...
0
votes
1answer
11 views

Definition of sigmoidal curve with epsilon

I want to create a sigmoidal curve $f(x)$ with the parameters $s$ and $\epsilon$ so that it has the following features: $f(0) = 0 +\epsilon$ $f(s) = 1 - \epsilon$ $f'(s/2)=1$ Is this possible? If ...
-1
votes
1answer
55 views

How to find the inverse of $f(x)=2x - x^2$? [on hold]

What is the inverse of $f(x)=2x - x^2$ in the domain (0,1) and the range (0,1)?
2
votes
3answers
55 views

What is the value of this function?

Consider the three-variable function defined at the following way for all natural numbers $n,x,y$ : $f(0,x,y) = x+y $ $f(n,x,0) = x$ $f(n,x,y) = f(n-1, $ $ $ $f(n,x,y-1) , $ $ $ $f(n,x,y-1)+y ) $. ...
0
votes
1answer
49 views

Functions with real domain but complex range, do they have any use?

For example if we define the square root function like this: $$\text{Sqrt}({x})= \begin{cases} \sqrt{x} & x\geq 0 \\ i\sqrt{-x} & x<0 \end{cases}$$ Or we could have an exponential ...
0
votes
1answer
19 views

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$ subject to $\int_\mathbb{R} f(x)\,dx = 1$.

Maximize $J[f] = \int_\mathbb{R} f(x)\log f(x)\,dx$ over smooth surjections $f : \mathbb{R}\to (0, \alpha)$, where $\alpha$ is a real number, subject to $\int_\mathbb{R} f(x)\,dx = 1$. I have no idea ...
3
votes
2answers
76 views

A ring is a connected set

I not know how to prove this: For example $$A=\{(x,y,z)\in \mathbb{R^3}\mid 1 < x^2 + y^2 + z^2<2 \}$$ I know that $$\partial A=S(0,1)\cup S(0, \sqrt{2})$$ can that help me at all? I was also ...
1
vote
1answer
50 views

Can you help me with a function with cos and radians?

I have this function: $$d(n)=15-2,5\cdot\cos\left(\pi\frac{n-31}{360}\right).$$ This function talks about the difference between when day start and day over. When I change $n$ for $5$ result for me ...
0
votes
0answers
14 views

Converting Properties to Specific Function

Is there an area in mathematics that deals with the idea of deriving functions from a given set of certain desired properties? The concept seems very important. You can read about the history of ...