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

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2
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0answers
26 views

2D Cauchy Distribution Peak [closed]

Is the general form of a 2D Cauchy Peak, if A is the amplitude: $$\frac{A}{1+\frac{(x-x_0)^2}{\gamma_x^2}+\frac{(y-y_0)^2}{\gamma_y^2}}$$ $?$
1
vote
2answers
41 views

Composition of a piecewise function and another function

I have this two functions. $f(x)=\arcsin \left(\dfrac{3-x}{3x-1} \right)$ and $g(x)=\begin{cases} 0 ;& |x| <\pi \\ \sin(2x);& |x| \ge \pi \end{cases}.$ I have to find $f \circ g$. I ...
0
votes
1answer
19 views

Exception to definition of a function.

My book gives me this definition of a function: A function $f$ is a special kind of relation,i.e $f\subset A\times B$,such that the following hold: for each $a\in A$ there exist $b \in B$ ...
1
vote
1answer
53 views

Functional inequality $f(x_1+x_2)\ge f(x_1)+f(x_2)$

Given a function $f$ on the interval $0\le x \le 1$. We know that this function is non-negative and $f(1)=1$. Moreover, for any two numbers $x_1$ and $x_2$ such that $x_1\ge 0, x_2 \ge 0$ and $x_1+x_2\...
0
votes
0answers
21 views

Iterating a relation to find a function

I was playing around with a graphing calculator, trying to find approximations for inverses of $f(x)=x^5+x+1$. This cannot be expressed with radicals or the like, but I wanted to see how close I could ...
3
votes
4answers
111 views

Disprove the statement $f(A \cap B) = f(A) \cap f(B)$ [duplicate]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: If $f : X \rightarrow Y$ is a function and $A$, $B$ are subsets of $X$ then $f(A \cap B) = f(A)...
1
vote
2answers
49 views

Recursively counting divisors of a number

I want to make a recursive function f that counts all (not only prime) different divisors of a given natural number: $f(n): = |{a ∈ ℕ | ∃ b ∈ ℕ : a . b = n }| $ ; with $ f(0)=0 $ for example $ f(3) ...
-3
votes
0answers
24 views

Prove a primitive recursive function

I'm supposed to prove that the division of integers (whole numbers) is primitive recursive. I know that the add, subs, mult are primitive recursive but I don't know if that helps, I'm not asking for ...
1
vote
1answer
25 views

The function $f:X\rightarrow X$ is defined in such a way: $f(x)=x-\frac1x, f^{(1)}(x)=f(x), f^{(n)}(x)=f\left(f^{(n-1)}(x)\right)$

Let the set $X=\mathbb Q / \{-1;0;1\}$. The function $f:X\rightarrow X$ is defined in such a way: $$f(x)=x-\frac1x, f^{(1)}(x)=f(x), f^{(n)}(x)=f\left(f^{(n-1)}(x)\right), n\in \mathbb N.$$ Is ...
0
votes
1answer
19 views

resolvable function

I need help. Show that the system of equations: $$y_1 + \cos(y_1y_2) = y_2*x_1 + 1$$ $$\sin(y_1) = x_2 + y_2$$ in an environment of $(x_1, x_2, y_1, y_2)=(0, −1, 0, 1)$ , $$\begin{pmatrix} y_1\\y_2 \...
0
votes
1answer
35 views

How can I find the largest possible subset A of $\mathbb{R}$?

So I have this equation $$f(x)= \frac{x^2}{(x-2)(x+3)}$$ and I need to find the largest possible subset $A$ of $\mathbb{R}$ that could form the domain of a function. Can anybody help me? I really don'...
2
votes
2answers
27 views

Complex function $f$ is either constant or unbounded, but maximum value still does exist even if $f$ is not constant?

In Complex Variables and Applications, Brown & Churchill (9th edition), I stumbled upon a chapter which got me somewhat confused. On page 175 of the book, there is the theorem, which states the ...
0
votes
0answers
20 views

Poll Ranking Formula

I am having a hard time making a ranking formula for a project I am working on. Here's a practical approach and I would really appreciate any help. Lets say we have $1000 to share among 5 people,...
0
votes
1answer
46 views

Simplification of this function

$f(x)= ((1- 4x^2)^{1/2} - 2(3)^{1/2}x)/((3-12x^2)^{1/2} + 2x)$ Find range when $x$ belongs to $(-(3)^1/2 /4 ,1/2)$ I have to find the range of this function , I have simplified this expression , I ...
2
votes
2answers
38 views

How do I find the domain/range of functions algebraically?

I've been having trouble when trying to find the domain/range of functions algebraically. Here is an example: $P(x)=\frac{1}{3+\sqrt{x+1}}$ Finding the domain: $x+1\ge0$ $x\ge-1$ Therefore, $x \...
0
votes
1answer
15 views

Finding Domain of a Function with a natural logarithm at the denominator of the fraction

I have the function: $y = f(x) = \frac{x}{\ln x}$ The function is undefined for the conditions: a denominator of a fraction being zero. a logarithm being negative or equal to zero. Hence, is ...
0
votes
1answer
28 views

Show that every local homeomorphism is continuous and open therefore bijective local homeomorphism is a homeomorphism

Follow up on another question I asked recently: Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism Definition: Let $(X, \mathcal{...
2
votes
2answers
40 views

Topology: Show restriction of continuous function is continuous, and restriction of a homeomorphism is a homeomorphism

I need to prove two trivial results but I don't know how to work with restricted function and its inverse Consider the topological spaces $(X, \mathcal{T}), (Y, \mathcal{J})$ Claim 1: Let $f:X \...
0
votes
3answers
87 views

Find the minimum of the value $n$ such that $(1-0.03)^n<0.03$

How can I find the smallest positive integer $n$ such that $$(1-0.03)^n<0.03$$ without the help of a computer?
38
votes
8answers
2k views

Does this pattern have anything to do with derivatives?

In 6th grade I was first introduced to the idea of a function in the form of tables. The input would be "n" and the output "$f_n$" would be some modification of the input. I remember finding a pattern ...
0
votes
2answers
46 views

Show that cuts are preserved under homeomorphism

Let $(X, \mathcal{T})$ be a topological space, assume that $X$ has no proper (not $X$ or $\varnothing$) clopen subset. Definition: A point $p \in X$ is a cut if $X \setminus\{p\}$ has a proper ...
2
votes
1answer
75 views

If $f(f(x)) = f(x^2)$, then must there be some constant $c$ such that $f(x)=c$ for all values of $x$ in the domain of $f$?

Here is a problem from Rusczyk-Crawford's Art of Problems Solving: Intermediate Algebra textbook (Chapter 2 Review, problem 2.30). If $f(f(x)) = f(x^2)$, then must there be some constant $c$ such ...
1
vote
1answer
31 views

Repeating/“Periodic” Derivatives? [duplicate]

We know that $Ce^x$ and $0$ are the two functions whose first derivative is equal to itself, but what about derivatives of a higher order? For example, the second derivative of $e^{-x}$ is equal to ...
0
votes
2answers
37 views

Cardinality of the Domain vs Codomain in Surjective (non-injective) & Injective (non-surjective) functions

I'm a student in college just beginning to study the basics of set theory. In studying about Surjective & Injective functions & how they map their domain to their codomain, it came to my mind ...
5
votes
2answers
56 views

Why isn't f(x^2) a horizontal stretch of f(x) by a factor of “1/x”?

I know this question seems silly, but it came to mind while reading about transforming functions. Is the statement "y=f(kx) results from scaling the graph of y=f(x) horizontally by a factor of 1/k" ...
0
votes
0answers
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
votes
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
votes
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(...
0
votes
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 $v_1=(1,2,0,3)$, $...
0
votes
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 ...
6
votes
4answers
125 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 ...
1
vote
1answer
12 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{ } ...
2
votes
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] \...
0
votes
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. ...
0
votes
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: $$\...
1
vote
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
votes
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
votes
2answers
46 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!
0
votes
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 ...
3
votes
5answers
146 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 ...
1
vote
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
votes
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}\...
0
votes
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 ...
3
votes
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,...
0
votes
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. $f(x,T)=\begin{cases} 1,x\geq T\\ 0, x<T \end{cases}$
2
votes
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 ...
-5
votes
1answer
36 views

Graph the function [closed]

Graph the function $$ f(x)=\begin{cases} -x-2, & -2<x\le -1 \\ -x^2, & -1<x\le 1 \\ x+2, & 1<x\le 2 \end{cases} $$
0
votes
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)...
0
votes
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
votes
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 ...