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

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2
votes
0answers
40 views

Real Analysis: Given $A_1$, $A_2$, $A_3$, which set is open, which is closed?

a. $A_1 = \{x: \sin(1/x) =0\}$ Clearly $\sin(1/x) = 0$ for $x = \frac{1}{n\pi}$, what can we say about this set? b. $A_2 = \{x: x\sin(1/x)= 0 \}$ Again $\sin(1/x) = 0$ for $x = \frac{1}{n\pi}$, is ...
-3
votes
1answer
16 views

Sketching the graph of a sequence of functions [on hold]

Does anyone know how to sketch graphs of a sequence of functions? Say, I need to sketch $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ for $x \in [0, 2]$. But I have no idea where to start. Can someone please help? ...
1
vote
0answers
8 views

baire $1$ function

Here is a new definition of Baire Class one function. Suppose that $X$ is a complete separable metric space. A function $f:X \rightarrow \mathbb{R}$ is said to be Baire class one if for any ...
0
votes
1answer
45 views

Find all functions $f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+$ such that $f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$

Find all functions $f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+$ such that $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ for all $x,y\in\mathbb{Q}^+$. Before this problem, I have solved few similar ...
6
votes
0answers
75 views

Can you add new functions to the set of elementary functions such that every function has an anti-derivative?

Its fairly well known that not every elementary function has an elementary anti-derivative. The common examples of this are $\exp(-x^2)$ and $\sin(x)/x$. The general workaround to this problem is to ...
-3
votes
0answers
44 views

Solving trigonometirc functions [closed]

I am having difficulty understanding this question. I have tried to find the solution but am coming up with a negative answer and the answer must be positive as t in this equation is time. I am not ...
0
votes
0answers
15 views

Bijective Function from n-fold Cartesian Product to set of functions

This is my first posting, so I apologize for the cumbersome formatting; I am not yet familiar with MathJax/Latex commands. Let $Y$ be a set and let $n$ be an element of $\mathbb N$ (natural numbers). ...
0
votes
1answer
30 views

How to solve for k to get equal roots in a composite function

I am given: $f(x) = 4x-2k\ $ and $\ g(x) = \dfrac{9}{2-x}$ ($x \not = 2$) The question is: Find the values for k for which the equation $fg(x) = x$ has two equal roots. So $f(g(x))=\dfrac{4\cdot ...
1
vote
2answers
47 views

Proving two functions are monotonically related

I've been relying on stackexchange a lot lately but this is my first time asking a question. A lot of searching has yielded no answer so hopefully someone can help out. I'm trying to find (and prove) ...
0
votes
1answer
54 views

Find the inverse function of $ f (x ) = x^2 - x - 2$

Find the inverse function of $ f (x ) = x^2 - x - 2$, where x is equal to or larger than 1/2. I tried to express it in form of $ (x - 1 )^2 = y + 2 $, but this is not true as the term in the middle ...
3
votes
1answer
32 views

biggest possible domain of diffeomorphism

Consider the function $$f:\mathbb{R}^2 \rightarrow \mathbb{R}^2, (x,y) \mapsto (x^2-y^2,2xy).$$ How can I determine a subset $D$ of $\mathbb{R}^2$ as big as possible such that $f|_D$ becomes a ...
0
votes
1answer
24 views

Don't understand this question [table of ordered pairs, find missing values]

I am very confused by this question that I have encountered while practicing for my GED. Over the last 6 months or so I've taken 3 official practice tests, and every time I took a test I encountered ...
4
votes
1answer
34 views

Let $\ f_1:A \rightarrow B$ and $\ f_2:A \rightarrow B$. Prove or disprove $f_1 \cap f_2$ iff $f_1=f_2$.

Here is the question I am working on (screenshot): So, I haven't worked with function proofs very much (especially in the context of iff statements and with intersections). I am looking to see ...
0
votes
1answer
14 views

Creating Polynomial Function with Surface Area of Cylinder

I've spent a few hours at this question but can't seem to get the right answer. I was hoping someone here can lead me in the right direction. The question: A storage tank is to be constructed ...
0
votes
1answer
30 views

Does the existence of a maximum over $|f_u(x,u(x))|$ imply that $f$ satisfies the Lipschitz condition?

A function $f$ satisfies the Lipschitz condition if: $$|f(x,u_1(x)) - f(x,u_2(x))| \leq A |u_1(x) - u_2(x)|$$ $$\frac {|f(x,u_1(x)) - f(x,u_2(x))|}{|u_1(x) - u_2(x)|} \leq A $$ Does the existence ...
-2
votes
0answers
57 views

What is this function? [closed]

I've seen some formula that seems to make start value approach to final value. Probably the formula is incorrect, but I believe that here, people will recognize what is that and write correct. Please, ...
2
votes
1answer
37 views

Substitution with functional equations

I've found this nice introduction worksheet that I started to work through with the goal to get a better understanding of functions and finding them in equations. I've gotten so far but in this one ...
0
votes
1answer
28 views

How to derive the general formula to determine the equation of a given cubic function

My question is: When determining the equation of a cubic function, we can separate the general cubic equation into it's solutions and we end up with the equation $y = a(x-r_1)(x-r_2)(x-r_3)$ We ...
0
votes
2answers
18 views

To find a extremal point of a function with parameters

I have a function $$f(x) = (x-5m)(x+m)^2$$ I have tried to find the extremal points of this function (and then find if it's local maxima or minima). That means I need to find the x of derivative. The ...
0
votes
0answers
20 views

Definition of positive definite function

I'm trying to understand a definition of a positive definite function I found in a book on Lyapunov stability Definition: A continuous function $W(x)$ is said to be a positive definite function if ...
0
votes
0answers
20 views

Derivative of a correlation function

From a big set of data I create a correlation function between a response parameter and three input parameters $(P_1, P_2, P_3)$. $Response = K_1 + K_2 \cdot P_1 + K_3 \cdot P_2 + K_4 \cdot P_3 + ...
0
votes
0answers
32 views

difference between limiting and special case

In Mathematics and Statistics we see generalized distributions having a number of parameters. varying the values of these parameters we get special or limiting distributions of the generalized ...
1
vote
1answer
36 views

Discrete mathematics, equivalence relations, functions.

I'd like some insight on how to 'solve' this problem (more towards understanding what the problem is asking) Suppose that $A$ is a nonempty set and $\mathcal{R}$ is an equivalence relation on ...
1
vote
1answer
33 views

Function Domain of cubic root [duplicate]

Does the domain of the function $y=\sqrt[3]{x^3+1}$ include $x<-1$? If yes, why is Mathematica and Wolfram Alpha not plotting that part of the function?
-4
votes
3answers
32 views

a question of mapping and set theory [closed]

A set $S$ is said to be infinite if there is a one-one correspondence between $S$ and any proper subset of $S$ prove The set of integers is infinite . The set of real number is infinite. If a set ...
1
vote
1answer
20 views

Fit Quantized Piecewise Constant Function to Another Piecewise Constant Function

I have a situation where I have a function $$f(x) : [r_1,r_2]\in\mathbb{R} \rightarrow [r_3,r_4]\in\mathbb{R}$$ and I need to fit a function $$g(x) : [r_1,r_2]\in\mathbb{R} \rightarrow ...
4
votes
1answer
59 views

If $h(x)=f(g(f(x)))$ is bijective, what do we know about $f,g$?

Question: If $h(x)=f(g(f(x)))$ as a function $\mathbb R \rightarrow \mathbb R$ is bijective, what do we know about $f,g$, which are also functions $\mathbb R \rightarrow \mathbb R$? Is my proof ...
1
vote
2answers
23 views

Inverse of homogenous function?

suppose that $\alpha>0$ and $f$ is an invertible function such that $f(\alpha x)=\alpha f(x)$. Does this automatically also imply that $f^{-1}(\alpha x)=\alpha f^{-1}(x)$? I would think yes ...
1
vote
4answers
85 views

Difference between $f(x(t))$ and $f(t,x)$

Why is there a difference between the two differential equations: $\overset{.}{x}(t)=f(x(t))$ and $\overset{.}{x}(t)=f(t,x)$ ?
-1
votes
0answers
16 views

What exactly are maps from $(I, \partial I) \to (I, \partial I) $?

The following is an excerpt from my textbook Let $\phi_1$ and $\phi_2$ be maps $(I, \partial I) \to (I, \partial I) $ What exactly are maps from $(I, \partial I) \to (I, \partial I) $? I don't ...
2
votes
2answers
38 views

Expressing every function as sum of an odd and an even function

If $f$ is to be written as a sum of the even function $E$ and the odd function $O$, $E=\dfrac{f(x) + f(-x)}{2} \quad$ and $O=\dfrac{f(x)-f(-x)}{2}$ obviously works. I get a bit confused though ...
1
vote
1answer
35 views

Define a relation — with functions and derivatives

Here is the problem I am working on: I am in a beginning level abstract math/proofs class, and haven't had much experience with calculus in any proof (or in any relation). Here is my understanding ...
1
vote
1answer
44 views

What does the sum of the reciprocals of composites run along?

This is fairly straight forward: $$\sum_{p\space\text{prime}}^x \frac{1}{p_x} \sim \ln(\ln(x))$$ And if $$\sum_{c\space \text{composite}}^x \frac{1}{c_x}\sim f(x)$$ Then what is $f(x)$?
-3
votes
2answers
32 views

exponential graphs of the form $y = B^x$ [closed]

What is true about 2 exponential graphs of the form $y=B^x$ if the value of $B$ in the first equation is n and the value of $B$ in the second equation is 1/n?
1
vote
0answers
40 views

Defining a function over time in terms of a random variable that is undefined at a certain time

Let $X_n$ be a random variable taking on one of three values $a,b$ or $c$ over time. That is, for each $n \in \mathbb{N}$, we have $X_n \in \{a,b,c\}$. Also, for each $n \in \mathbb{N}$, let $F_n$ be ...
-1
votes
0answers
15 views

Graphing Functions Help [closed]

What is the same for all exponential equations of the form $y=B^x$ if: A) The value of $B$ is greater than $1$? b) The value pf $B$ is greater than $0$ and less than $1$?
2
votes
1answer
37 views

Inverse and composite functions [closed]

If $f(x)=\frac{x}{1-√x}$, $x≥0$ and $g(x)=3x+1, $ Solve the equation $(f^{-1}\circ g)(x)=9/16$. Hint:do not attempt to find $f^{-1}(x)$.
3
votes
5answers
94 views

Difference between $f(f(x)) = f(x)$ and $f(x) = x$?

So I don't seem to have understood the concept of a function. There are three similar problems and it was on the third problem that I noticed I did not reason correctly, but I don't know why: a) ...
0
votes
1answer
13 views

Construct an exactly smooth function as a cutoff of half ball with vanishing normal dirivative

$\newcommand{\pt}{\partial}$ Suppose $B_r^+:=\{x=(x_1,x_2)\in B_r(0)\subset R^2|x_2\geq0\}$, can we construct a $C^\infty$ smooth function $\phi$, $0\leq\phi\leq1$, such that $$ \phi\equiv1 \text{ in ...
-1
votes
2answers
51 views

Why the length of the zigzag curve approximating the circle does not approach the length of the circle?

I recently bumped into this question which asks why $\pi=4$ is wrong. And some answers(see the answer of user TCL, for example) stated that this has to do with functions and their derivatives. ...
0
votes
1answer
40 views

Convergent conjecture: What is the proof?

Lets say that $\def\nn{\mathbb{N}}$$\def\rr{\mathbb{R}}$$K : \nn \to \rr$ and $\displaystyle \sum_{i=1}^\infty \frac{K(i)}{K(i+1)}$ is a convergent sum. My conjecture is that the function $K$ must be ...
-4
votes
0answers
45 views

Second derivative at 0. [closed]

Here is an $f(x)=((x-2)^2-2)^2-2)^2-\cdots))))\cdots)$ $n$ times. I have to calculate the second derivative at 0. I solved it in a very complicated way, I’m sure there’s a better solution.
0
votes
1answer
14 views

How to find singular points of a function without knowing the graph?

Problem: Let $f(x) = (x-1)^{2/3} - (x+1)^{2/3}$. Locate and classify all local extreme values of this function. Determine whether any of these extreme values are absolute. Attempt at solution: We ...
0
votes
2answers
27 views

Function in Maple giving weird plots

In Maple I have defined a function If I plot within $Q \in [0,100]$ I get but I get the exact same plot with other boundaries and if I use $Q \in [0, 10^{-10}]$ I get How can this happen? ...
2
votes
1answer
36 views

How to find this kind of function?

I am trying to find a function $f(x,y)$ with $f:\mathbb{R}^{2} \rightarrow [a, b]$ where $[a, b]$ is equal to $[-1, 1]$, $[0, 1]$, or some other small interval (open intervals are fine as well). The ...
3
votes
2answers
81 views

Finding $F(x)$ from $F(kx),$ where $F(x)$ is the antiderivative of the function $f(x)$.

I have that $F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1$, and I would like to find $F(x)$. Attempt Since $F(e^{x}x) = e^{x}x^{2} - e^{x}x + e^{x} - 1,$ $F(t) = \alpha_{1}t^{\beta_{1}} + ...
-2
votes
0answers
31 views

Functions with infinite ranges [closed]

Let $Y$ be an infinite subset of a set $X$. Let $f\colon X \to Y$ and $g\colon X \to Y$ be two functions from $X$ into $Y$ with cardinal numbers of ranges of $f$ and $g$ equal to the cardinal number ...
2
votes
1answer
56 views

Solving the functional equation $2f(x)-f(1/x)=3x$

If $$2f(x)-f(1/x)=3x$$ how would I find $f(x)$? I have tried various linear and other functions but I do not know how to start this
5
votes
0answers
123 views

Apartness of reals and algorithm exctraction

I am trying to wrap my head around the notion of apartness in constructive mathematics and it turns out I lack understanding miserably. I would like to use as elementary notions as possible, in the ...
-1
votes
1answer
34 views

bounded & Lipschtiz function--> bounded & uniformly continuous function [closed]

if $f\in C([0,T]\times \mathbb{R})$ is a bounded and uniformly continuous function, how can we construct a sequence of bounded and Lipschtiz function $\{f_n\}$ such that $$f_n\to f, \quad ...