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

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1answer
45 views

Find $P(12)$ when $P(x)={1\over(x+1)}$, for $x=0,1,2,…11$

let $P(x)$ be a polynomial of degree $11$ such that $P(x)={1\over(x+1)}$, for $x=0,1,2,......11$ then find value of $P(12)$
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1answer
15 views

a functions for positive integers

For a modeling, i need a descending function. for example i need something like this: ...
2
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1answer
41 views

What is the range of that function?

What is the range of the function $$f(x,y,z):=\left \{\frac{xyz}{xy+xz+yz} \right \}$$ over all the natural numbers $x,\,y,\,z$ (Zero does not belong to the naturals.), where $\{x\}$ stands for the ...
0
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1answer
43 views

Periodicity and period of a function

The question is : Let $f(x)$ be a real valued function defined for all real numbers x such that for for some fixed real number $a>0$, $f(x+a)=\frac{1}{2} + \sqrt{f(x)-(f(x))^2}$ and $\frac12\le ...
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1answer
25 views

Function $f\colon R\to N, x\mapsto \sqrt{x}$ Find domain of :D

Function $$f\colon R\longrightarrow N, x\longmapsto \sqrt{x}$$ Find domain of :D Help me please :D
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2answers
24 views

finding a differentiable function

$f(x,y)$ is a differentiable function satisfying the following properties: $f(x+t, y)= f(x,y) + ty$ and $f(x, y+t)= f(x,y) + tx$, $\forall x, y, t \in\mathbb{R}$ and $f(z, 0) = k$ for any ...
2
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2answers
49 views

Is $\sec^{-1}(\sec(\pi/2)) = \pi/2$?

I think it shouldn't be defined as $\pi/2$ is not in the range of the function $\sec^{-1}(x)$ Wolfram confused me by giving the answer as $\pi/2$ : Link But it mentions on another page that $\pi/2$ ...
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1answer
44 views

Defining a bijective function from $2\mathbb{N}$ to $3\mathbb{Z}-1$?

$2\mathbb{N}=\{2n:n\in\mathbb{N}\}$ and $3\mathbb{Z}-1=\{3n-1:n\in\mathbb{Z}\}$ Work: So far, my plan is to first define a bijective function from $2\mathbb{N}$ to $\mathbb{N}$ and then define ...
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2answers
14 views

Percentage lost

I need a formula to calculate something like this if I have 5000 dollars and loose 5% of it per DAY. How many days is it going to take to have 10 dollars left in my pockets.
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3answers
58 views

Proving that function $f:[0,\infty)\rightarrow [0,\infty)$ defined by $f(x)=\frac{x^2}{1-x}$ is bijective.

I am having a bit of trouble with the algebra for proving that the function is injective. Basically I set $f(a)=f(b)$ for $a,b\in[0,\infty)$ and $a,b\neq 1$. ...
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2answers
50 views

Something not working out for me in the continuity definition

I'm studying analysis and I've ran into this proposition saying that a function from a metric space X to a metric space Y, is continuous if and only if for every open set O in Y, the inverse image of ...
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0answers
53 views

How do I go about showing the cardinality of two sets are the same?

How do I go about showing that the cardinality of the set of natural numbers and the cardinality of the cartesian product of integers is the same?: |N|=|Z x Z| Directly |N| = Aleph-null and I can ...
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1answer
45 views

Metric space, inequation

I have a question to the following problem: Let $(X,d)$ be a metrical space and let be $f:[0,\infty)\to[0,\infty)$ twice differentiable with $f(0)=0$, $f(x)>0$ for $x>0$, $f'\geq 0$ and ...
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1answer
19 views

Establishing convexity of a function

Let $\theta \in \Theta \subset \mathbb{R}^k$. I have the following objective function $$ F(\theta):=||\max(0,f_1(\theta)),...,\max(0,f_n(\theta))||^2 $$ where $||\cdot||$ is the Euclidean Norm and ...
0
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1answer
42 views

Basic Function Question… regarding codomain and range?

I'm trying to figure out the answer to this question and I think I have an answer, I just don t know if I'm right. Would you guys mind helping me out? ...
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3answers
71 views

Equation $2f(x+2)+3f(x-2)=x+5$ Find $f(1)$ [closed]

Here is equation : $$2f(x+2)+3f(x-2)=x+5$$ How to find $f(1)$ ?
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2answers
96 views

Integrable functions and absolute values

I have qutoted that the absolute value of an integral is less than or equal to the integral of an absolute value of a function. I have also said $|-g(x)| \le g(x) \le |g(x)|$ implies the integral ...
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2answers
39 views

How to find a function from an infinite sequence of derivatives at $x=0$

I need an odd function $f(x)$ which converges to $\pm \infty$ at $\pm a$ for some positive $a$. At $x=0$, the even derivatives must be $0$, and the odd derivatives must be factorials : $f(0)=0$, ...
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0answers
30 views

Creating non-linear function to calculate points per distance in reversed order.

I have a small game in which I want to give points according to closeness to location, so the maximum points will be given for the minimal number = 1. It's similar to GeoGuesser game. Here is the data ...
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1answer
25 views

Setting a function in function of another one

So I have H(E,X)=0.01EX and g(x)=0.02[x-0.001x^2) where G=H , so I want to redo the whole thing so everything is function of E, the result should be something like that Y(E)= 10E-0.4E^2 soo..how do ...
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1answer
33 views

Prove Every Function is a Relation

In my notes my professor has a question to prove that $\forall m,n \in \Bbb N^+$, $2^{mn}\ge n^m$. There is a suggestion that it can be proved by taking the logarithm of the inequality so that $mn ...
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0answers
29 views

Formula for calculation score based on distance

I try to write function for calculating scoring from distance in my game. I found something similar : link But I need the distance(x) to be between 0-31855000 meters and score between 0 to 1000. ...
2
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3answers
33 views

Show that $x_0$ must be an integer. Conclude that $\sqrt[n]{2}$ is irrational for every $n \geq 2$

I have a problem in my workbook that is as follows: Let $f = x^n + a_{n-1}x^{n-1}+\dots+a_1x+a_0 = 0 $ with $a_i \in \mathbb{Z}$. Suppose there exists a rational number $x_0$ with $f(x_0) = 0$. ...
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1answer
22 views

Graphing derivatives of non-function equations

Is it possible to graph the derivatives of equations that fail the vertical line test? Such as a circle, a folium of descartes, an asteroid, etc?
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0answers
39 views

What mathematical function can give me a set of curves similar to these?

I'm looking for a set of functions that can give me curves similar to these. Any ideas? The red circle was drawed just as a guide.
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1answer
50 views

finding all functions: $f(\frac{x+a}{b})=f(\frac{f(x)+a}{b})$,$f:\mathbb{Q}\rightarrow \mathbb{Z}$

If i have this equation: $f(\frac{x+a}{b})=f(\frac{f(x)+a}{b})$ such that $x\in \mathbb{Q}$, $a\in \mathbb{Z}$ , $b\in\mathbb{N}$ and $f:\mathbb{Q}\rightarrow \mathbb{Z}$ Need to find all functions ...
2
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0answers
141 views

Problem solution by model theory

Sorry if that's not the right place for asking this, but didn't have anywhere else to go. I was cheking out some math problems in the Mathematical Olympiad site, and I found this one: Let $\mathbb ...
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1answer
166 views

Beginner proof of image of functions and functions of sets

This is the third time I got my proofs handed back from my teacher. She won't tell me what's wrong except I have to redo it. I am running out of luck and I need help towards the right direction! The ...
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1answer
89 views

Let $X$ be any set and $M=\{\emptyset, X\}$. Prove that the class of measurable functions are exactly those functions that are constant on X.

I am attempting to solve a suggested problem while studying for my upcoming real analysis exam. Could somebody please help me with this question? Question: Let $X$ be any set and $M=\{\emptyset , ...
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1answer
44 views

find the maximum of the function F under the condition $ \sum_{i=1}^N x_i = 1$

Let F a function of $ \mathbb{R} ^N_+ \rightarrow \mathbb{R}$ defined as : $$F(x_1,..,x_N)= - \sum_{i=1}^N x_i log(x_i) , x_i \gt 0$$ How can i find the maximum of the function F under the ...
2
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0answers
23 views

Finding expression that runs through zeros of a function

A list plot of the zeros of the imaginary part of $2^s\pi^{s-1}\sin\bigg(\frac{\pi s}{2}\bigg)\Gamma(1-s)$ for $s=\frac{1}{2}+it$ looks like this: How would I find the expression for the line ...
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1answer
31 views

Finding domains of functions

I'm doing college algebra homework and I always think that if something seems too easy, you're probably wrong, so I wanted to check. The domain of a function is all real numbers unless it creates a ...
2
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1answer
44 views

what are all the functions that satisfying $f(\frac{x+a}{b})=f(\frac{f(x)+a}{b})$

Given $f(\frac{x+a}{b})=f(\frac{f(x)+a}{b})$, $x$ is a real number, $a$ is an integer number and $b$ is a natural number. What are all the functions that satisfying this restriction? I tried to ...
8
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1answer
99 views

A question of rationality

This problem was asked to me by a friend and I simply have no idea about it. So I have not progressed a single bit. The problem is this: If $f :\mathbb{R}\to \mathbb{R}$ is an infinitely ...
2
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3answers
964 views

“nice functions”

I see the statement of "nice functions" in textbooks and the authors usually don't need to give the definition of "nice functions". For example in a book which I read now the authors write "Morrey ...
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0answers
37 views

Is this functional linear?

I know it's trivial, but is this functional not linear? $\phi:\mathbb{R}[X]\ni p \rightarrow p(0)p \in \mathbb{R}[X]$ $$\phi(p+q)=(p+q)(0)\cdot(p+q)=(p(0)+q(0))\cdot(p+q)\ne\phi(p)+\phi(q)$$
3
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0answers
50 views

How to approach sketching sine and cosine graphs with transformations

Any tips or suggestions in sketching these graphs quickly, and in ONE go? In exams, I don't want to spend ages re-drawing the original sine/cosine graph, one by one, following each new ...
1
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1answer
28 views

Continuity of the operator

Let $D: (C^1[a,b], \Vert .\Vert_1 )\rightarrow (C[a,b], \Vert .\Vert_1)$ with $D(f)=f^{\prime}$ I wonder if I will be continuing this operator Note: $\Vert f\Vert_1=\int_a^b\vert f(x)\vert dx$ All ...
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0answers
45 views

Matlab functions of variables

So I am writing a function to compute the following equations for an SIR model: So here's my code: ...
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3answers
131 views

Examples of one-to-one and onto functions $f:\mathbb N\to \mathbb N$

Give examples of functions from $\mathbb{N}$ to $\mathbb{N}$ with the following properties: i. one-to-one but not onto ii. onto but not one-to-one iii. both onto and ...
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1answer
40 views

Expression as argument in function definition

When a function definition has an expression (instead of just a single variable) as the argument to the function, what does this mean? For example, I have this question (part b): Given a certain ...
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1answer
365 views

Matlab using subscript variable

I'm trying to write a function in matlab but I don't quite know if it is working. In the equation line i have: xdot(2) = N_h * x; to signify: $$\frac{dy}{dt} = ...
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1answer
165 views

Let f : X → Y and g : Y → Z be bijective mappings. Show that gf is bijective

Let f : X → Y and g : Y → Z be bijective mappings. Show that gf, the composition of f and g, is bijective. I have that since f(x)=y, and g(y)=z we get g(f(x))=g(y)=z is this enough to show gf is ...
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1answer
1k views

Series expansion of square root function

Could I please know how to expand: $$\sqrt{4-3x^2}$$ I simplified it to $\sqrt{1-\frac34x^2}$ but the question is if I let $x = x^2$, what will the coefficient of, say $x^5$ or $x^{10}$ be in the ...
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2answers
155 views

Solving complex trig functions: $\sin2x + \sin3x = \frac{\sqrt{3}}2$

How to solve: $$\sin(2x) + \sin(3x) = \frac{\sqrt{3}}{2}$$ where $x$ is in $[-\pi,\pi]$? I have no idea what to do with the $\sin(2x) + \sin(3x)$. Am I supposed to factorise, differentiate, is ...
2
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1answer
58 views

Show that $S^1 - \lbrace (1,0)\rbrace$ is homeomorphic to the open interval $(0,1)$

Be $S^1$ the unit circle in the plane, that is, $S^1= \lbrace (x,y) : x^2+y^2=1 \rbrace$ with the subspace topology. Show that $S^1 - \lbrace (1,0)\rbrace$ is homeomorphic to the open interval ...
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1answer
50 views

Riemann sum of $\sin x$ on [0,1]

For each positive integer $n$, define a function $f_n$ on [0,1] as follows: $f_n(x)$= $0$ $\forall$ $x=0$ $\sin$($\pi\over{2n}$) $\forall x\in(0,{1\over{n}}]$ $\sin$($2\pi\over{2n}$) $\forall ...
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2answers
35 views

Bijection of a function.

Define the function f: $(2,\infty) -> (-\infty,-1)$ by $f(x)= \frac{-x}{x-2}$. Show that f is bijective. I know i need to prove both injective and surjective, and I was able to solve the equation ...
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1answer
60 views

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2)

Let P(x) be a polynomial of degree 4 , having extremum at $x=1,x=2$ and $\lim_{x\to 0}\frac{x^2+P(x)}{x^2}=2$ Then the value of P(2) is _______ I worked out the limit using L'Hospital got a relation ...
2
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3answers
142 views

Range of f(x) = $\frac{\sqrt3\,\sin x}{2 + \cos x}$ [duplicate]

Can you give any idea about the range of the following function? $$f(x) = \frac{\sqrt{3}\,\sin x}{2 + \cos x}$$