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

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3
votes
1answer
33 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
1answer
38 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
25 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
33 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
38 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
33 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
21 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
61 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
36 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
45 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 $f(x)$?
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 ...
2
votes
1answer
38 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
14 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 ...
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
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
132 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
vote
1answer
44 views

Prove that $C(m,n)=\frac{(m+n)(m+n+1)}{2}+m$ is a bijection from $\mathbb{N}^2$ to $\mathbb{N}$ [duplicate]

I've been stuck for a couple of hours on how to prove that $C(m,n)=\frac{(m+n)(m+n+1)}{2}+m$ is a bijection from $\mathbb{N}^2$ to $\mathbb{N}$. I read in another question that in order to prove that ...
0
votes
1answer
43 views

Understanding $F \circ \phi^{-1}$ in differential geometry

I am struggling with a question in elementary differential geometry. I thought I understood the basics until I read page 20 of The Geometry of Physics by T. Frankel. Suppose we have a manifold of ...
0
votes
0answers
27 views

higher order derivatives of three composite functions

How can I obtain a formula for higher order derivatives for composite of three functions as $f(g(h(x)))$?
2
votes
3answers
54 views

How find minimum $f(x) = \max_{t\in[-1,1]} \left| t+ \frac{3}{2+t} + x \right|$

How find minimum this function $f(x) = \max_{t\in[-1,1]} \left| t+ \frac{3}{2+t} + x \right|$?
1
vote
3answers
62 views

Finding the lowest value y can have in $y = \frac{1}{2}(e^x - e^{-x}) + \frac{n}{2}(e^x + e^{-x }) $

How can I find the lowest value $y$ can have when $n$ is greater than or equal to $2$ using only algebra? $$y = \frac{1}{2}(e^x - e^{-x}) + \frac{n}{2}(e^x + e^{-x })$$
-1
votes
1answer
54 views

Result related to $f (x+y+z) =f (x)f (y)f (z)$ [closed]

If $f (x+y+z) =f (x)f (y)f (z) $for all real $x,y,z$ and $f(2)=4$ and $f'(0)=3$. Then, how to find $f(0)$ and $f'(2)$?
0
votes
2answers
41 views

find the range of $f(x)=\sqrt{x^2−9}$

$$f(x)=\sqrt{x^2−9}$$ I know that the domain of square root is greater than or equal to zero. I solve for when $x^2−9\ge 0$ and get $x^2\ge9$. Now I get $x\ge 3$ and x≤−3. So that the domain would be ...
1
vote
5answers
134 views

Specific integral creating a constant

Let's say: $${\int_{n}^{n+\frac{1}{n}}f(x) }\space\text {d}x=C$$ I am looking for some function $f$ that would create, for all $n$ values inputted, a constant $C$ to be created. What is $f(x)$? What ...
0
votes
3answers
47 views

Solving rational inequalities

I am having difficulty with solving: $$\frac{(2x-1)}{(x-5)} > \frac{(x+1)}{(x+5)}.$$ I tried to move the one side over so that there was zero opposite the equation but I ended up making some ...
-2
votes
1answer
30 views

Transformations to obtain a new equation. [closed]

Another question from my online course that I don't understand. Describe how the graph of $y=-2f(3(x-1))-4$ can be obtained from the graph of $f(x) = x^4$. I am not sure what the $f$ value is or ...
1
vote
1answer
77 views

Need to prove continuous periodic function of $\varphi (x) \equiv \psi(x)$

Question: Let two $\varphi(x) $ and $\psi(x)$ periodic and continous functions such that $$ \lim_{ x\to\infty}(\varphi(x)-\psi(x))=0, \quad x\in \mathbb{R}. $$ Prove that $$ ...
1
vote
1answer
38 views

Incorrect proposal to dealing with nested radicals?

Lets try to calculate this set of nested radicals: $$f(2)=\sqrt{2\sqrt{3\cdots}}$$ So If we call: $$f(n)=\sqrt{n\sqrt{(n+1)\sqrt{(n+2)\cdots}}}$$ Of course, this is for: $f(2)$ right, so logically ...
4
votes
1answer
95 views

Vieta's Formula failed?

Find the value of $p$ if $p$ and $q$ are the roots of the equation, $x^2+px+q=0, \ \ \{x,p,q\}\in\ \mathbb{R}$ By using vieta's formula for sum and product of roots, $\begin{cases} p+q=-p ...
1
vote
2answers
29 views

Mooculus beginner explanation if f(-1) = -7 and f(x) = g(-x*6), what point satisfies?

I have been looking at this for awhile and I am having a mental block maybe. The last question of the first section of the Mooculus book is If $f(-1) = -7$ and $f(x) = g(-6x)$, what point must ...
-1
votes
2answers
37 views

Determining remainders when dividing polynomials.

In my online course I also have to determine the remainder of this equation and I am stuck on it as we have never divided a polynomial by anything larger than (x-a) variables. Determine the remainder ...
0
votes
2answers
18 views

Help with function substitution problem!

Question: If $f(x) = x^3-(a+b)x^2 + abx$, find the value of $f(a)$ and explain the significance of $(x – a)$. Im doing a grade $12$ advanced functions course online at the moment and this is ...