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

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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
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 ...
-5
votes
0answers
46 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
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 + ...
-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, ...
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
24 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
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 ...
2
votes
2answers
463 views

Comparing and contrasting equations and functions

I have several related questions, so I'm going to label them to make sure I understand what questions that answers are referring to. I understand that a function is an expression that produces one ...
1
vote
5answers
133 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 ...
1
vote
1answer
37 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?
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
60 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 ...
5
votes
4answers
32k views

How to prove if a function is bijective?

I am having problems being able to formally demonstrate when a function is bijective (and therefore, surjective and injective). Here's an example: How do I prove that $g(x)$ is bijective? ...
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
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 ...
5
votes
3answers
145 views

Can we obtain $f(y+x)=y+f(x)$ from $f(x^2+f(x)^2+x)=f(x)^2+x^2+f(x)$?

Find all function $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that $$f(m^2+f(n))=f(m)^2+n.$$ Let $P(x,y)$ be the assertion: $f(x^2+f(y))=f(x)^2+y \; \forall x,y \in \mathbb{Z}^+.$ $P(x,x)$ ...
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 $\overset{.}{x}(t)=f(t,x)$ ?
5
votes
0answers
131 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 ...
5
votes
1answer
215 views

How to compare the similarity between functions?

I'm designing a web service that finds the regression function of a pattern within an image. I analyzed three images and found the following three regressions: 1) $f(x) = 74.7602 + 0.2005x - ...
3
votes
1answer
64 views

How many such functions are possible?

Let $f$ be a function from $\{1,2,3,\dots,10\}$ to $\Bbb{R}$ such that ...
-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 ...
3
votes
1answer
720 views

How to prove a function is periodic?

$$f(x)=\begin{cases}1&\text{if }2n-1<x<2n,\\0&\text{if }2n<x<2n+1. \end{cases} $$ Is this function is periodic or not? How can I prove it?
-2
votes
1answer
66 views

Continuous logic function

I'm trying to find a continuous function $f:[0,1]\times [0,1] \to \{0,1\}$, so that on the sub-domains $[0,1/2), [1/2,1]$ $f$ behaves like an AND gate (or OR gate). I don't think it can be done, ...
-1
votes
2answers
84 views

The range of the function $f(x,y)=(x+y,xy)$

I have the following homework question: $$\begin{split} f: \mathbb I \times \mathbb I &\to \mathbb R\times \mathbb R\\ f(x, y) &=(x+y, xy)\end{split}$$ Does there exist $(x, y) \in \mathbb ...
1
vote
1answer
46 views

Easily computable function that is one-to-one and onto with 2 or more inputs

I am looking for a function that will take take n inputs and have 1 output, and knowing the 1 output and function you can get back the n inputs, without error and computationally easy in both ...
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 ...
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
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 $f(x)$?
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 $f^{-1}(x)$.
0
votes
2answers
23 views

Understanding functions of matrices

Given $$f(X) = rank(X) $$ with X being a matrix. Is it possible to visualize such a function? What is the space that it lives in (assuming all entries live in $\mathbb{R}$)? Is there literature on ...
0
votes
2answers
301 views

Proving that hyperbolic sinh is bijective

I have to prove that sinh is bijective. So first i try to profe that it is 1-1: $f(a)= f(b) => a=b$ I will use proof by contradiction: let f(a)= f(b) and $a$ doesn't equal to $b$. After i ...
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) ...
1
vote
3answers
53 views

What is the limit of $\lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $ [closed]

How do i calculate the limit of this function? $$ \lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $$ I have no idea where to start.
-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
1k views

How to combine an amount of money with the compound interest function?

Tommy has some money at home from his graduation modeled by the function $h(x)=350$. He read about a bank that has savings accounts that accrue interest according to the function $s(x)= 1.04 ...
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 ...
-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 $f'(2)$?
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}} + ...
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 ...
2
votes
3answers
130 views

$\sin x + x\cos x=0$ solution?

Any idea of solving this equation? $$\sin x + x\cos x=0$$ I have also tried by setting a function $g(x)=\sin x+x\cos x$ and searching for solutions using the derivative but my atempts w
-4
votes
3answers
45 views

For what values of k is $y = kx^2 - 4x - 2$ a negative definite [closed]

For what values of $k$ is $$ y = kx^2 - 4x - 2 $$ a negative definite? I know what the discriminant is, but I just don't know what to do with it. Not exactly sure what a negative definite ...
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
3
votes
2answers
83 views

Are there some functions that cannot be optimized using calculus?

I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical ...
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
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 ...