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

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58 views

Example of a bijection from the set of real numbers to a subset of irrationals

I need an example of a bijection from the set of real numbers to a subset of the irrationals. I tried something like $f(x)=x+\sqrt{2}$, but where should I map $-\sqrt{2}$?
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3answers
43 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
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1answer
22 views

Analyzing a particular type of functions

Let $f$ be a function from $\mathbb{Z}$ to $ \mathbb{Z}$ Now $f(x)=x$ Question: Is $f$ continuous in its domain?( perhaps yes by epsilon delta argument but I don't know if I am justified in doing ...
0
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2answers
31 views

maps from infinite sets to infinite sets

I know that the set of irrationals is uncountable, but I feel that there can always a bijection from one infinite set to a subset of another infinite set. Does this sound right? Say, from Irrationals ...
1
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1answer
33 views

Evaluate position of first secondary maximum of $\frac{\sin N (x/2)}{\sin (x/2)}$

The function $$f(x) = \displaystyle \left | \frac{\sin \left( N \displaystyle \frac{x}{2} \right)}{\sin \left( \displaystyle \frac{x}{2} \right)} \right |$$ when evaluated for $x > 0$, has its ...
2
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1answer
43 views

Does something that is injective, surjective or bijective imply that it is a function?

As the title says. Sorry it seems like a silly question but it's something I've been wondering because it seems like sometimes the word "function" is omitted, but other times it is included
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1answer
51 views

Which function should I choose?

I am writing a paper and looking for a function.Its shape is just like this: when its x value is very small ,its y is close to 0 , but when x value is a little big, then its y value is very close to ...
0
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3answers
35 views

Function with limit constraints

In working with predator-prey population modeling, I ended up requiring a function $f(x)$ that satisfies the following conditions: $f$ is real and continuous, and so is $\frac{df}{dx}$; ...
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0answers
34 views

Is there a name for this property of a real function?

Let $M=\sup_{x \in [0,1]^n} f(x)$ where $f:[0,1]^n \rightarrow \mathbb{R}$ is differentiable twice, and write $x=(x_1, \dots, x_n)$. Let $M_{x_i=0}=\sup_{x \in [0,1]^n:x_i=0} f(x)$ and ...
0
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0answers
38 views

L'Hospital's rule for higher derivatives

Let $u,v \in C^\infty(\mathbb{R})$, where $u(0) = 0$ and $v(0) = 0$ and $v'(0) \not= 0$. Then, one can define a function $f \in C^\infty(\mathbb{R}\setminus\{0\})$ by $f := u/v$. L'Hospital allows ...
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3answers
42 views

Finding domain of $f\text{ o }g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then ...
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3answers
81 views

If $H = \{(1,-4),(2,-3),(3,-2),(4,-1),…\},\;$ what is $H(9)$? [closed]

If $H = \{(1,-4),(2,-3),(3,-2),(4,-1),...\}$, what is $H(9)$? This is the problem on our lesson about relation and functions. Please help.
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0answers
35 views

Derive $f(Ax)=f(x)g(A)$: the property for scale invariance.

This is part of a proof for the following question: Show that if a probability density function $f(x)$ with $x>0$ is scale invariant then $f(Ax)=\frac{1}Af(x)$ where $A$ is a real constant. ...
2
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3answers
114 views

Finding the maximum and minimum values of $f(x)=a^x+a^{1/x}$

Let $f(x)=a^x+a^{1/x}\ (x\gt 0)$ where $a\in\mathbb R$ is a constant. Question 1 : What is the maximum value of $f(x)$ for $0\lt a\lt 1$? Question 2 : What is the minimum value of $f(x)$ for ...
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2answers
31 views

Calculate random integer inside a range of real numbers

$$F : \Bbb R \times \Bbb R \rightarrow \Bbb N $$ $$F(\text{minReal},\ \text{maxReal}) = \text{randomInt} \in \left[\text{minReal},\ \text{maxReal}\right] $$ Let $r \in [0, 1)$ be a random value. How ...
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0answers
27 views

How do points change in a curved surface?

In the middle picture it shows a row of sticks at certain points along a flat surface. Now in the outer left picture (never-mind the outer right one), when the surface becomes curved the points ...
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1answer
40 views

How to derive this formula about the bracket function?

Is there a direct way of proving that $$ [nx] = [x] + [x+\frac{1}{2}] + [x+\frac{1}{3}] + \ldots + [x+ \frac{1}{n}]$$ for each real number $x$ and for each positive integer $n$? My effort: Let ...
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3answers
43 views

Help with function proof [closed]

I am asked to prove or disprove that if $f:A\rightarrow B$ is a function, then: If $Y\subseteq B$, then $f^{-1}(B\setminus Y) = f^{-1}(B)\setminus f^{-1}(Y)$. I have no idea how to go about doing ...
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5answers
129 views

How prove $f(x)$ is a monotonic function if $f(x+y)=f(x)f(y)$

Let $f(x)$ be a real valued, differentiable function such that for any $x,y \in \mathbb{R}$,$f(x+y)=f(x)f(y)$. Suppose there exist $a,b$ such that $f(a)\neq 0, f'(b)>0$. Show that $f(x)$ ...
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2answers
43 views

showing the function is continuous at a point using $\epsilon$ and $\delta$

I have this question: Use the definition of continuous function with $\epsilon$ and $\delta$ to show that the function $f$, defined as $$f(x)=\begin{cases}0&\textrm{if } x=0 \\x \sin\frac{1}x ...
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2answers
98 views

Finding a root of a function via Rolle's theorem

Consider the function $f(t)=a(1-t)\cos(at)-\sin (at)$, where $a\in\mathbb R$. To show that it has a root in the unit interval I am urged to integrate $f$ and apply Rolle's Theorem. Attempt: $$\int ...
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2answers
35 views

Converting from set of Cartesian equations to Polar Equation

Is it possible to convert the set of Cartesian equations: $$x(t) = (20-30)*\cos(2t)+45*\cos(2t*(20-30)/20))$$ $$y(t) = (20-30)*\sin(2t)+45*\sin(2t*(20-30)/20))$$ where $$t \in [0,2\pi)$$ Into a ...
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2answers
36 views

Getting a function constant values from it's f(x)

Hello I'm new to this StackExchange site, in case this is off topic please point me out in the right direction. It's been about 3 years since I solved a math problem so I need some directions (not ...
0
votes
4answers
30 views

Empty preimage of an intersection implies empty intersection of the preimages

Assume $f:A\to A'$ is a function, $B\subset A'$, $C\subset A'$, and $f^{-1}(B\cap C)=\emptyset$ How can we see that $f^{-1}(B)\cap f^{-1}(C)=\emptyset$?
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2answers
37 views

Proof when the circle map is ergodic

Let $E=[0,1)$ with Lebesgue measure. For $a \in E$ consider the mapping $\theta_a:E \rightarrow E, \ \ \theta_a(x) = (x+a) \mod \ 1$. a) Show that $\theta_a$ is not ergodic when $a$ is rational. ...
1
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1answer
13 views

Creating switches for piece wise defined function?

How can I create "switches" [the term may be new, but I'll explain it] for piecewise defined functions ? Suppose a function: $$ f(x)=\begin{cases}\alpha\;,x\in D_1\\\beta\;,x\in ...
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0answers
33 views

Limit of measurable function is measurable

This question has been asked already here but I didn't get a satisfactory solution and didn't want to bring up an old question. Here is the question : Let $\{f_n\}$ be a sequence of measurable ...
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0answers
28 views

Functional characterization of zeroth law of thermodynamics [Sepration of Variables]

Zeroth law of thermodynamics is stated also as: If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C. This can be formulated ...
1
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1answer
21 views

Definition of the Domain of a Function when the sets are the elements

In case I have a function that calculate the normalized distance of elements in two sets $A$ and $B$ I can define the function as $\mathrm{elementDistance} : A \times B \rightarrow [0,1]$. But if I ...
1
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2answers
59 views

If a uniquness for all functions exist shouldn't there be uniquness to recursion?

What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example. Let's make a table of values for $^{x}2$ (0,1) (1,2) (2,4) ...
1
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1answer
30 views

Laplace Transform with sin and cos

Hi I am having trouble figuring out the solution of this Laplace transform: $$L_t{(u(t- \pi)(2\cos(t)-3\sin(3t))}$$ Where I am stuck if I am even on the right track is: $$L_t{(u(t- ...
1
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1answer
19 views

How $n^d \times m([0, \frac{1}{n}[^d) = m([0, 1[^d)$ follows from translation invariance and (finite) additivity

In this StackExchange question (which itself seems to reference to an exercise in Terence Tao's lecture notes on introductory measure theory on his blog here), it's said that assuming "finite ...
3
votes
1answer
31 views

Non-monotonic functions on ordered sets

I'm trying to prove that if $~~(A,<_A)~~$ and $~~(B,<_B)~~$ are linearly ordered sets and $~~f: A \rightarrow B~~$ is non-monotonic function than there exist points $~~a,b,c\in A~~$ such that ...
0
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1answer
42 views

How to identify the function of a graph?

I am trying to identify the function of a graph in order to create a dataset for it. Dataset as in several x/y-values that will lead Excel/PowerPoint to create a graph looking just like the drawing: ...
0
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1answer
24 views

Dirac delta function and well behaved function [duplicate]

whether dirac delta function a well behaved function? Can u please explain the properties of a well behaved function..?
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1answer
31 views

Find domain of function with quadratic numerator algebraically

I'm stuck on this problem: $$f(x) = \frac{x^2 -4}{x}$$ I need to determine why this function's domain is not: $$\{x|x \neq \pm 2\}$$ All of the examples that I've seen have the quadratic in the ...
0
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1answer
27 views

Need help creating a special function

I'm creating a special function in a game and needed some help with the maths end of it. Essentially, I need a programmable, non-linear function so that $f(100) = 0$, and $f(0) = 100$ (or some other ...
0
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1answer
24 views

Solving function in difference quotien equation

I have the problem Find the difference quotient $\frac{f(2 + h) - f(2)}{h}$ for $f(x) = \frac{1}{x^2}$. The answer they gave is $\frac{-(4 + h)}{4(2 + h)^2}$ So far I've done: $$\frac{[1/(2 + h)^2 ...
3
votes
1answer
78 views

Is $f(t)=(\cos(t),\sin(t))$ a function?

In the Linear Algebra book we're using (Linear Algebra with Applications, Bretscher, p.129), the author defines this as the function of the unit circle. I understand why the equation of a circle ...
1
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3answers
62 views

Show that $x^3 +x-1$ has a zero between $x=0$ and $x=1$

Show that $x^3 +x-1$ has a zero between $x=0$ and $x=1$, does anyone know how to go about starting this problem? I am basically clueless. I thought maybe at first polynomial division since its $x^3$, ...
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1answer
53 views

Find $k$ so that $f(x)$ is a continuous function [closed]

Find $k$ so that $f(x)$ is a continuous function. $$f(x)=\left\{\begin{array}{ll}x^2 &x\leq2\\ k-x^2 & x>2 \end{array}\right.$$ Does anyone know how to go about this problem? Thanks in ...
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0answers
7 views

Proper name for the problem (finding optimal discrete function)

Given a set $D = \{d_1, d_2, ..., d_N\}$, a set of some subsets of $D$, $D^\ast$ and a set of classes, $C = \{c_1, c_2, ..., c_M\}$, I want to find function, that maps a sequence $({d_i}_1^\ast, ...
1
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2answers
46 views

To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$

If $f: \mathbb R \to \mathbb R$ be a function such that for some $n_o \in \mathbb N$ , the $n_o$th iterate of $f$ has a unique fixed point $b$ , then how to prove that $f(b)=b$ ? I cant think of ...
0
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1answer
29 views

Existence of function satisfying given conditions?

Let $f:[0,1]\longrightarrow[0,1]$ be continuous, strictly increasing and $f(1)=1$. Suppose further that $f(x)>x$ for all $x\in[0,1)$. Is there any function satisfying the above conditions? My ...
2
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1answer
49 views

Let $f$ be continuous and $U \subset \mathbb{R}^n$ open, if $f: U \rightarrow \mathbb{R}^m $ is injective then $n \leq m$?

I had intended to restrict the image then $f:U \rightarrow f(U) \subset \mathbb{R}^m $ is bijective. Therefore $\dim f(U) = n \leq m$. That's right?
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0answers
42 views

A continuous differentiable map of R to (0;1)

Is there a single, continuously differentiable function $g(x,k)$ that approximates the following: $f(x)= \begin{cases} 0 & x<0 \\ x & 0 \le x \le 1 \\ 1 & x>1\end{cases}$ $k$ is a ...
1
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3answers
70 views

Why do we define functions to be set theoretic objects?

Why do we define functions to be set theoretic objects? Functions are so intuitive, why do we define it in complicated set theory language?
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3answers
252 views

Why is this proof of $\mathbb{N}\times\mathbb{N}$ being countable not formal?

My copy of Introduction to Real Analysis: Bartle and Sherbert gives: Theorem: The set $\mathbb{N}\times\mathbb{N}$ is countable. Informal Proof: Recall that $\mathbb{N}\times\mathbb{N}$ ...
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5answers
172 views

How to prepare this function for integration

I want to prepare $$f(x)=\frac{x}{1+x^2}$$ for integration, how do i get the $1+x^2$ to the top? Is $$\frac{x}{1+x^2}$$ the same as $\frac x1 + \frac{x}{x^2}$? If not please explain how I prepare the ...
0
votes
2answers
21 views

Getting to answer on difference quotient/function problem

Q: Find the difference quotient $\dfrac{f(x) - f(3)}{x - 3}$ for $f(x) = \dfrac{1}{x}$ Ans a: $\dfrac{1}{3x}$ Haven't been able to get to that answer. I got the bottom $3x$ right once but the top ...