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

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20 views

Determining bounds for a sum with nested infinite series

I am computing the inner product of the characters of the trivial and the $k$-th irreducible two dimensional representations of the dihedral group $D_n$ of order $2 n$ when $n$ is even. The ...
2
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2answers
55 views

How to show that the following function is bijective?

If we have the function $c : \mathbb{N}^2 \rightarrow \mathbb{N} : (x,y) \rightarrow 2^x \cdot (2y+1) -1 $ how to show that this function is bijective? So I thought the easiest way is to show that is ...
2
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3answers
47 views

Functions validity.

Why does writing a function differently make it valid for a originally invalid input? $e.g:$ $$f(x) = \frac{1} {(\frac1x+2)(\frac1x-3)} \implies x≠0$$ Which may alternatively be written as: $$f(...
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0answers
21 views

Finding a basis and the dimenson of $Im(f)$ and $ker(f)$.

$B_1=(p_1(x),p_2(x),p_3(x),p_4(x))$ , basis of $R_3[x]$. $B_2=(v_1,v_2,v_3,v_4)$, basis of $R^4.$ $p_1(x)=1+x$, $p_2(x)=1-x+x^2,$ $p_3(x)=x-2x^3,$ $p_4(x)=3+x^3,$ and $v_1=(1,2,0,3)$, $...
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1answer
11 views

Modelate a function based on imputs

I am making a game( nothing too fancy) and the game is consisting of many levels each with an unique identifier, a number $n\in\mathbb{N};6\leq n \leq 29$. Based on that identifier I need to place an ...
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4answers
127 views

The function $(-1)^{-x}$

I was bored so I put functions in Wolfram Alpha. And I got something that looks like a sin function. And in addition to that, the real part was continuous and the imaginary part was a cos function. It ...
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1answer
13 views

Functions composition commutativity

I have to prove that $\circ$ is not, in general, a commutative operation of Funct(X,X). My approach: Let X be a set, $a,b\in X$, $a\neq b$ constants. Let $i,j \in Funct(X,X)$ with $i:X \to X,\text{ } ...
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2answers
53 views

Preimage of a function

I'm having difficulties with the notion of preimage, specifically with this example: Let $A$ be a subset of $[0, 1]$. We define $$f(x) = \begin{cases} x, & x \in A; \\ -x, & x \in [0, 1] \...
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1answer
19 views

Find the Domain of the Function g such that gf and g inverse exists.

The functions $f$ and $g$ are defined as follows: $$f(x)=(2e^x-1)^2+2, x\in\mathbb R$$ $$g(x)=(x-1)^2-1, x\ge k$$ Find the range of values of $k$ such that both functions $gf$ and $g^{-1}$ exists. ...
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1answer
34 views

Solutions of a system of polynomial equations

I am trying to find the critical points of some functions such as $$f(x, y) = x^4 − x^2y^2 + y^3 − 18x^2 + 3y^2$$ I calculate the gradient, and then find a system of polynomial equations: $$\...
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1answer
31 views

Recursive to non recursive function

$$ f(x) = \begin{cases} 0 & x=1 \\ f(x-1)+1 & \frac{f(x-1)}{x-1} < p \\ f(x-1) & \text{otherwise} \\ \end{cases} $$ Where $p$ is a constant less that or equal to 1. And x is a whole ...
2
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2answers
55 views

Are we suposed to consider $f(0)=1$?

If you plot the graph of $f(x)=\frac{\sin x}{x}$ on GeoGebra, you would realise that the function is shown to be continuous at $x=0$. I do agree the $\lim_\limits{x\to 0}\frac{\sin x}{x}=1$, but $\...
2
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2answers
47 views

Is $ f \circ g $ invertible in the diagram below?

I was working through Can the composition of two non-invertible functions be invertible? For the image below is $f \circ g$ invertible? Thanks!
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2answers
85 views

Is the function continuous and differentiable in $\mathbb{R}$?

$$g(x)=\begin{cases} 3x-3 & x \leq 1 \\ 5x^2+2x-7 & x>1 \end{cases}$$ How do I determine if this function continuous and differentiable in $\mathbb{R}$? I know the solutions, but I don't ...
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5answers
147 views

$f(x) = 0$ when $x$ is $0$, and $1$ otherwise

I've been trying to create a function that will return $0$ when $x$ is $0$, and for any other $x$ value it should return $1$. I've searched for a pre-existing function online too and wasn't able to ...
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2answers
47 views

Is it correct to say $\text{max}(x_1,x_2,\dots,x_n)=\lim_{p \to \infty} \frac{x_1^p+x_2^p+\dots+x_n^p}{x_1^{p-1}+x_2^{p-1}+\dots+x_n^{p-1}}$

There are many possible way to represent the maximum function, I came up with this one: $$\text{max}(x_1,x_2,\dots,x_n)=\lim_{p \to \infty} \frac{x_1^p+x_2^p+\dots+x_n^p}{x_1^{p-1}+x_2^{p-1}+\dots+...
4
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1answer
64 views

if the equation $(x-2)e^x+a(x-1)^2=0$ have two real roots,Prove $a>0$

if the equation $$(x-2)e^x+a(x-1)^2=0,x\in R$$ have two real roots. show that $$a>0$$ Following is a solution since $$-a=\dfrac{(x-2)e^x}{(x-1)^2}$$ Let $$g(x)=\dfrac{(x-2)e^x}{(x-1)^2}\...
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1answer
49 views

Show that … Has No Real Roots

When $f(x) = 3x-4$ and $g(x) = \frac{5}{3-x}$, Question 1: Find the value of x for which fg(x) = 5 Question 2: Show that the equation $f^{-1}(x) = g^{-1}(x)$ has no real roots. I understand that ...
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3answers
78 views

Proving that a function is surjective

I want to prove that the function $\mathbb{N}_0 \times \mathbb{N}_0 \rightarrow \mathbb{N}_0$ defined as $(x,y) \mapsto 2^x \cdot (2y + 1) - 1$ is bijective. I have already proven that it is injective,...
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1answer
14 views

How to design a threshold function without using any comparison operator?

What are some methods to design a function that outputs $1$ if the input value $x$ is greater than a threshold $T$ and $0$ otherwise. $f(x,T)=\begin{cases} 1,x\geq T\\ 0, x<T \end{cases}$
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1answer
53 views

Is it possible to construct such a function in analytical form?

Suppose $f\left(f\left(x\right)\right)=\sin(x)$ Is it possible to find $f$ in closed form, or any other forms so as to visualize $f(x)$ on $x\in[-\pi,\pi]$? Is it possible to prove the existence and ...
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1answer
36 views

Graph the function [closed]

Graph the function $$ f(x)=\begin{cases} -x-2, & -2<x\le -1 \\ -x^2, & -1<x\le 1 \\ x+2, & 1<x\le 2 \end{cases} $$
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1answer
32 views

Precalculus questions: Domain, range, and composition of functions

Directions: evaluate each of the functions at the indicated value of $x$. construct each of the functions, then find the domain. If $f(x)=\{(3,5),(2,4),(1,7)\}$, $g(x)=\sqrt{x-3}$, $h(x)=\{(3,2),(4,3)...
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1answer
37 views

Proof of a Surjective Function

I've run into a question in my textbook and I'm not sure if I understand fully the answer from the solution manual. Here is the question: Problem: Suppose that $f: A \rightarrow B$ is any function. ...
0
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2answers
50 views

Find a function that is a bijection $f:(0,1) \rightarrow (1, \infty)$

Find a function that is a bijection $f:(0,1) \rightarrow (1, \infty)$ I am to assume the intervals have the same cardinality. I honestly don't even know how to begin with this. Can you provide me ...
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0answers
17 views

Image maps for Relations

So I noticed a rather interesting thing about the image function and preimage function. The wikipedia page for Image: https://en.wikipedia.org/wiki/Image_(mathematics) (notation for image section) ...
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3answers
71 views

How is the function $f: \mathbb{Z} \to \mathbb{R}$ continuous?

Where $\mathbb{Z}$ is the set of integers and $\mathbb{R}$ the set of real numbers. In a question in a problem sheet, it said this statement was correct, however I do not understand how. You ...
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2answers
23 views

Description of preimage of $g$ defined by $g|_A = f|_A, g(S \setminus A) = \{x\}$

$S$ is a given directed set. $X$ is a first countable topological space. There is a function $f : S\rightarrow X$. With help of this $f$ , a new function $g:S\rightarrow X$ is being defined like the ...
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0answers
28 views

How do I determine Taylor polynomial of degree 2 of function?

How do I determine Taylor polynomial of degree 2 of function $g(x,y) = (x^2 + y)e^{xy}$ in development at point $(x_0,y_0)=(1,-2)$? There is this question, but the problem is that I need to do it on ...
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1answer
22 views

How do I determine set and sketch image of function $g(\mathbb{R}^2)$ if $g(x,y)=\begin{pmatrix} e^x\cos y\\ e^x\sin y\end{pmatrix}$?

How do I determine set and sketch image of function $g(\mathbb{R}^2)$ if $g(x,y)=\begin{pmatrix} e^x\cos y\\ e^x\sin y\end{pmatrix}$?
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1answer
34 views

Find functions $F(\mathbf{x})$ invariant under a map $\mathbf{x} \to \mathbf{x'}$

We introduce a map $\mathbf{x} \to \mathbf{x'}$, defined as (for example on $\mathbb{R}^3$): $$x'=f(x,y,z) \\ y'=g(x,y,z) \\ z'=h(x,y,z)$$ Note that $f,g,h$ are not all linear (or at least, I'm not ...
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1answer
36 views

How to solve problem like this - even & odd function

This is my problem, Consider $f(x)=\frac{1}{e^{x}}$ , Where $x\epsilon [0,\infty[$ 1) Define the function $g:\Re \rightarrow \Re$ such that $g$ is an even function and $g=f$ on $[0,\infty[$ Some ...
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0answers
26 views

Inequality between norm of function and it's derivative

There is a theorem: Let $f$ be a continuously differentiable, $2\pi$-periodic function. Given $\int_{-\pi}^{\pi} f(x) dx = 0$, I need to prove that $$||f|| \le \frac{\pi}{2} \cdot ||f'||.$$ Where ...
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1answer
44 views

How to sketch an image and find set of this function?

$ c\left(t\right) =\begin{pmatrix} e^{t} \\ e^{-t} \end{pmatrix} $ How do I find set and geometric object of $ c\left(\mathbb R \right) $ ? I know the graph of $e^{t}$ and $e^{-t}$, but how to put ...
0
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1answer
26 views

How do I determine some function if domain and range are already specified?

In one of my college interviews, I was asked to define a function whose domain is (0,1) and range is (-1,1). This can be very well answered (we can modify 'x' in cos(x)). I did encounter certain ...
3
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3answers
30 views

reciprocal factor of absolute value when evaluating a square root expression

Learning with an old russian math book, i found the following evaluation for the function $f(x)=\sqrt{1+x^2}$: $f(\frac1x)=\vert x \vert^{-1}\sqrt{1+x^2}$ My evaluation gave me $\sqrt{1+\dfrac1{x^...
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0answers
28 views

Function which gives number of primes STRICTLY less than a number x.

We all know that the $\pi(x)$ function (Prime-counting function) gives us the number of primes less than or equal to x. I'm interested in the function which specifically gives the number of primes ...
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1answer
21 views

Proving the complete additivity of the Big Omega function $\Omega(n)$ (total number of prime factors of n) .

The Big Omega function $\Omega(n)$ gives you the total number of prime factors of the number n. A function $f(x)$ is completely additive if $f(ab)=f(a)+f(b)$ for all positive numbers $a$ and $b$, ...
2
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0answers
27 views

Find the sets for $f:R \rightarrow R$ given by $f(x) = \left|x\right|$

Find the sets for $f:R \rightarrow R$ given by $f(x) = \left|x\right|$. Let $S= [0,4]$ and $T = [-3,0]$. Find the sets: $f(S)$, $f(T)$, $f(S) \cap f(T)$, and $f(S \cap T)$. Is $f(S) \cap f(T) = f(...
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2answers
52 views

How do I determine the maximum or the minimum of in the range the function in the range $B=\left \{ \left ( x,y \right ):x^2+y^2\leq 1 \right \}$?

How do I determine the maximum or the minimum of in the range the function $f(x,y)=x^3+y^3 $in the range $B=\left \{ \left ( x,y \right ):x^2+y^2\leq 1 \right \}$? As a continuous function f must ...
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0answers
21 views

Symmetry vs. commutativity and more..

I was reading the following segment of an article about commutativity: ...
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0answers
42 views

My proof that $f^{-1}(D_1 \cap D_2) = f^{-1}(D_1) \cap f^{-1}(D_2)$

I'm currently self studying proof and set-theory, and I'm quite new to both of them. As an exercise, I'm practicing proving some basic theorems, so it'll be great if you can give me some feedback on ...
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3answers
28 views

Find the inverse of $f(x) = 1 + \frac{1}{x}, x \gt 0$

I'm tasked to find the inverse of the function $$f(x) = 1 + \frac{1}{x}, x \gt 0$$ The book offers a solution, simply to set $$1 + \frac{1}{x} = s$$ and solve $$x = \frac{1}{s-1}$$ and I think I ...
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1answer
44 views

Injectivity in functions

Sorry, I know that it has to be a very simple problem, but I'm frustrated because of it. Let $f,g:\mathbb{N}^3→\mathbb{N}f$: $f(x,y,z)=3^x⋅5^y⋅7^z$ and $g(x,y,z)=3^x+5^y+7^z$ Prove that: $1.f$ is ...
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1answer
29 views

$c_1\cosh(x)+c_2\sinh(x)=A\cosh(x+y)$ always true?

My question: Can I rewrite $c_1\cosh(x)+c_2\sinh(x)$, which is a solution to a differential equation as $$A\cosh(x+x_0)$$ introducing the new constants of integration $A$ and $x_0$? How can I deal ...
0
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1answer
39 views

Determine and sketch the image of function?

I have to sketch the image of function $h\left(\mathbb R^{2} \right)$ as 1. a set and as 2. a geometric object. $h\left(r,\phi\right)=\begin{pmatrix} rcos\phi \\ rsin\phi \\ r \end{pmatrix} $ I ...
1
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2answers
68 views

Finding all possible values of a Function

Let a function be defined as $f:N\to N$ and $x-f(x)= 19\left[\dfrac{x}{19}\right] - 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N$ and $1900<f(1990)<2000$. Find all values of $f(1990)$. $...
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0answers
42 views

Surface area of a bottle with integration

Would it be possible to model a bottle using a function, then revolving it to determine the surface area and the volume while customizing the curvature and the dimensions of particular sections of the ...
0
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0answers
15 views

show that Lb(a, y) := max{1−ay−by,0}, (a, y) ∈ R×{−1,1}, b ∈ R, is continuouse whit respect to first variable

Show that: $$L_b(a, y) = \max\{1−ay−by,0\},\;\; (a, y) \in\mathbb{R}×\{−1,1\}, b\in\mathbb{R}$$ is continuous whit respect to first variable.
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1answer
15 views

Find the first derivative of given limit

Let $f(x)$ is a polynomial satisfying $f(x).f(y)=f(x) + f(y) +f(xy) -2 $ for all x ,y and $f(2)=1025$ , then the value of lim x tending to 2 $f'(x)$ is I want to know that value at $f(1)=1$ can ...