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

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0answers
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6
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4answers
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How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
0
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1answer
33 views

Approximate function as $x$ tends to infinity

I'm looking for a way to approximate the following function $f$ as $x \to \infty$ $$ f = \ln \left( 1 + e^{a_1 x} + e^{a_2 x} + A e^{(a_1+a_2) x} \right) $$ where $a_1$, $a_2$ and $A$ are constants. ...
0
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3answers
33 views

Why ordered sequences can be reduced to sets?

I am trying to understand why ordered sequences can be reduced to basic sets. I understand most of the following proof: Sequences can be defined as functions Functions are a special case of ...
0
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2answers
27 views

Drawing squares of functions

If I want to sketch the square of the sinc function, or any function for that matter, is there a neat transformation technique which would allow one not to refer to graphing devices for this task?
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3answers
46 views

Find the maximum of a non-linear function with 4 parameters

I'm trying to find the maximum of a function with 4 positive parameters : $$f(x,y,z,t)=$$$$(-2(x+5)^2+200x)+(-2(y+10)^2+200y)+(-2(z+15)^2+200z)+(-2t^2+200t)$$ with $x+y+z+t = 150$ I don't know if ...
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0answers
29 views

Algebra 2 help!!?… [on hold]

For the function below, find the difference quotient f(x)-f(a)/x-a and then simplify. f(x) =6x-5
1
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3answers
55 views

Pigeonhole principle 3

I need help on this question, I'm lost and really don't know how to proceed: Use the pigeonhole principle to prove that in a round-robin chess tournament (with 18 participants) there will be at least ...
0
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2answers
25 views

Find the maximum of a function with 4 parameters

I'm trying to find the maximum of a function with 4 positive parameters : $$f(x,y,z,t)=(2x+2)+(4y-1)+(3z+4)+(5t+3)$$ with $x+y+z+t = 50$ I don't know if this is feasible and how to proceed. I have ...
2
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1answer
34 views

General formula for a specific problem?

I have a problem which I would like to have a general formula for. Here is the description. There are island aligned by a grid. Every cell contains an island. Every adjacent island is connected by ...
0
votes
1answer
22 views

Find functions that satisfy this equation.

Give some examples of functions, $F$ and $G$ such that $$x=\sqrt{F(x)+G(x)\sqrt{F(x+n)}}-\sqrt{F(x+n)}.$$ $n$ can be a constant. [Edit]: with $n\gt{0}$
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1answer
47 views

Function with increasing property.

Prove that $\frac{1}{2}(x+2)^{-3/2}-(\frac{1}{2}x+3)(x+3)^{-3/2}$ is increasing function for $x\ge4$. I tried it by taking its first derivative but by first derivative for me its difficult to say it ...
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2answers
32 views

Generalization of Cantor Pairing function to triples and n-tuples

Is there a generalization for the Cantor Pairing function to (ordered) triples and ultimately to (ordered) n-tuples? It's however important that the there exists an inverse function: computing z from ...
1
vote
2answers
53 views

Why is there no inverse function in this case?

There is no inverse function for $R(q) = 40q - 4q^2$ for $0≤q≤10$ But when $0≤q≤5$ there is inverse function. It's something about the function being one-to-one but I don't know why it isn't ...
1
vote
1answer
33 views

Is this relation symmetric and transitive?

Set A is given as $A = \{1,2,3,4,5,6,7,8,9,10,11,13,14\} $ And is defined as $R = \{(x,y) : 3x = y\}$ The relation that I'm getting is: $ R = \{(3,3), (6,6), (9,9), (12,12)\} $ Over here, it is ...
0
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1answer
6 views

If PQis a focal chord, show that the interval RU is parallel to the axis of the parabola.

For part (c) of question thirteen am I only required to find the gradient of RU and prove that is it zero? This is how I have interpreted this question. ANY help on the matter is much appreciated ...
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0answers
11 views

Constructing a function [on hold]

The revenue R that a store earns from selling a particular item is given by the product of the number of units sold x, and the price per unit p. Construct the revenue function for the price function ...
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2answers
33 views

Finding the correct slope.

To determine the slope of the graph of this relation do I take the two points as (4,20), (0,0) and then proceed to take 20-0=20 and 4-0=4, to divide 20 by 4 to get the slope of 5m? For the ...
-4
votes
2answers
59 views

How to simplify $(x+1) / (x^3-x)$ [on hold]

For all $x$ in the domain of the function $\frac{x+1}{x^3-x}$, this function is equivalent to which of the following? (A) $\dfrac{1}{x^2}-\dfrac{1}{x^3}$ (B) ...
0
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1answer
14 views

how many matches are played

so the question is the are a total of 16 teams. Each team will play each other once so each team will play 15 matches since they can't play themselves. How many matches in total are played?
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3answers
87 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 ?
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3answers
38 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 ...
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2answers
50 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) ...
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5answers
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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 ...
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3answers
785 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 ...
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1answer
42 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 $$B=\{b_1,b_2,…,b_m\}$$ ...
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2answers
33 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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1answer
29 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. ...
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3answers
87 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 $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
2
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1answer
45 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
44 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
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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
115 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 ...
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0answers
32 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 ...
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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
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0answers
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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
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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
21 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
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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 ...
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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
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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
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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
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5answers
124 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
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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
28 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
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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
47 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
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1answer
31 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 ...