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

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7
votes
4answers
63 views

Showing a function $f: \mathbb{N} \times\mathbb{N} \to \mathbb{N}$ is injective

Let $f: \mathbb{N} \times\mathbb{N} \to \mathbb{N}$ with $$ f(i,j) = \frac{(i+j-2)(i+j-1)}{2}+j. $$ I want to show $f$ is an injection. This is how I approached the problem: I tried to show ...
2
votes
1answer
26 views

Does the concept of “cograph of a function” have natural generalisations / extensions?

First, definitions: The graph of a function $f : A \to B$ is a subset of $A \times B$, namely the set $\{(x,y) : x \in A, y \in B, f(x) = y\}$. The cograph of a function $f : A \to B$ is the ...
0
votes
1answer
41 views

Find the range of this function [on hold]

How do you find the range of the following function, please? $$\frac{2x^2 + 20}{x^2 + 5}$$
0
votes
0answers
12 views

Complexity of computing a posiform of a quadratic pseudo-boolean function

I am reading the chapter 13, Pseudo-Boolean functions, of Boolean Functions: Theory, Algorithms, and Applications by Crama et. al. In section 13.2, the authors introduce the idea of Posiform. The ...
-3
votes
2answers
32 views

Find the rang of $\sin (a) + \sin (b)$ [on hold]

If : $a+b=\frac{\pi }{2}$, Find the range of $$\sin (a) + \sin (b)$$
1
vote
3answers
39 views

Evaluating a function at a point where $x =$ matrix.

Given $A=\left( \begin{array} {lcr} 1 & -1\\ 2 & 3 \end{array} \right)$ and $f(x)=x^2-3x+3$ calculate $f(A)$. I tried to consider the constant $3$ as $3$ times the identity matrix ($3I$) but ...
0
votes
0answers
41 views

Is it possible to define an inverse of the main three trig. functions without domain restrictions?

Ok, I know that the main three main trigonometric functions, that is the tangent, sine, and cosine, are periodic and thus not one-to-one, but onto. And, since an inverse requires a function to be onto ...
0
votes
1answer
33 views

Explain why this composite function is not allowed?

Explain why this composite function is not allowed when $f(x) = 2x+1, x \in [-5,5]$ and $g(x) = x^2, x \in \mathbb{R}, x \geq 0$ How would you change the domains so that the function $fg(x)$ can ...
4
votes
1answer
72 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: $f(x^2+f(y))=f(x)^2+y \; \forall x,y \in \mathbb{Z}^+.$ $P(x,x)$ ...
-2
votes
0answers
44 views

How to prove that composition of functions is a function [on hold]

Using the fact that a function is a relation, which is a subset of the product of $X$ and $Y$. $(a,b)$ belongs to $f$ and $(a,c)$ belongs to $f \implies b=c$
1
vote
0answers
54 views

Confused about basic of image

Hello I tried to work a problem from the text called " Introduction to Real Analysis" by Robert G Bartle and Donald Sherbert and I encountered a small difficulty. I am starting to think that my ...
3
votes
4answers
49 views

What does $f^{-1}(B)= \{ x \in X \mid f(x) \in B\}$ mean?

I have encountered the expression $$f^{-1}(B) = \{ x \in X \mid f(x) \in B\}$$ My questions are: 1) What does the $-1$ exponent mean in this context? 2) Is it right to say "if the set $X$ ...
4
votes
2answers
32 views

Prove that $f(X\cap f^{-1}(Y))=f(X)\cap Y$

Let $\ f\colon A\to B$ and let $X\subset A$, $Y\subset B$, prove that $$f(X\cap f^{-1}(Y))=f(X)\cap Y$$ The "$\subset$"$-$inclusion is easy: if $y\in f(X\cap f^{-1}(Y))$, exists a $x\in X\cap ...
2
votes
2answers
66 views

What is the actual definition of a function?

I am learning precalculus and my book defines the following: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to every element $a$ in $A$ one and only one value in $B$. Well, I ...
0
votes
1answer
54 views

What is the function $f$ such that $\sum_{k=0}^n f(k)=n^3$?

$$\begin{align*} 1 &\leadsto 1 \\ 1+3 &\leadsto 2^2 \\ 1+3+5 &\leadsto 3^2 \end{align*}$$ In general, if $f(x)=2x+1$, then $f(0)+f(1)+f(2)...f(n)=(n+1)^2$. Now, $$\begin{align*} 1 ...
1
vote
1answer
17 views

Find formula structure for a complex function

I am looking to find the function formula structure of a repeating function like the one in the image linked below.... Something that repeats indefinitely (like a sine wave) on the X-axis. Anybody ...
0
votes
0answers
27 views
0
votes
1answer
41 views

Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and $h=f-g$

Please please please please please I want some help ,Is there and body here who can help me in this question : Let $(X,Σ,μ)$ be a measure space and $f$ and $g$ are positive integrable functions and ...
3
votes
1answer
243 views

If the derivative is zero on [a, b] so the function is constant - using Heine-Borel?

I know the proof using MVT but I was wondering if it can be proofed using Heine-Borel Lemma, that "Every open cover of close interval has a finite subcover". (without compactness, simple as that). ...
0
votes
0answers
24 views

Piecewise Logistic Function [Satellite Data]

I am working with $16$-day MODIS EVI (satellite) data and I want to fit a Piece-wise Logistic Function through my $23$ EVI data values. The following formula is for the Piece-wise Logistic Function: ...
-1
votes
2answers
61 views

Let $f(x)$ be a polynomial such that $f(a)=b, f(b)=c, f(c)=a$ Then Prove that $a=b=c$. [on hold]

Let $f(x)$ be a polynomial in $x$ With integer coefficient. If for natural numbers $a,b,c$, $f(a)=b, f(b)=c, f(c)=a$ Prove that $a=b=c$.
0
votes
1answer
34 views

well defined mapping-function

I would like to know how to show an mapping or function is well defined i think in generale we use that : -$f$ is well defined mapping iff $( x\in E\implies f(x)\in F)$ in particular when mapping ...
1
vote
2answers
43 views

Is 'a' differentiable in f when f is a product of a differentiable and non-differentiable function?

Recently, I was studying differentiable and non-differentiable functions and I wondered whether this "conjecture" of mine is true: 1) "If $f(x)$ is a function that is the product of $g(x)$ and $h(x)$ ...
1
vote
3answers
46 views

$f(f(y)+1)=y+f(1)$ is bijective.

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(xf(y)+x)=xy+f(x), \; \forall x,y \in \mathbb{R}.$$ I read a solution in finding this function. It states that setting $x=1$ ...
1
vote
2answers
22 views

Can we say that a function is increasing/decreasing on some range if there's a vertical asymptote in that range?

The graph below shows the function $f(x)=\frac{e^x}{x-1}$ Can we say that the function is decreasing for all $x\le2$ (there's a local minimum at $x=2$) or do we have to take the asymptote at $x=1$ ...
2
votes
1answer
30 views

open problems regarding functions

I am looking for some open problems regarding functions. Problems like, Whether a function satisfying some properties say, X,Y,Z, exists or not, is unknown. Like there is no function $f(x)$ such ...
0
votes
1answer
23 views

Counting functions and stirling numbers

Let S= { f | f: A $\rightarrow$ B, |Image(f)|=k}. |A|=m, |B|=n. where k $ \le n, k \le m $ |S|=$ {n \choose k} $ S(m,k) k!. where S(m,k) are the striling numbers of the second kind. What I can't ...
1
vote
1answer
74 views

Is there a name for the function of a semicircle?

Recently I've learned many different names for different types of functions... but I've been wondering, is there a name for this type of function? $\sqrt{x - x^2}$
3
votes
0answers
27 views

Technical name for an almost-monotonic function

I'm wondering if there’s a technical name or short phrase that describes a function that’s monotonic, subject to some uniformly bounded amount of backtracking. $\exists \epsilon \forall x , y : y \gt ...
0
votes
4answers
48 views

Must a continuous function on $\mathbb R$ with only rational values be constant? [duplicate]

As I'm preparing for my exam I have to solve the following question: Determine if the following is correct: Let $f$ be a continuous function is $\Bbb R$. If $f$ recieves only rational values, ...
3
votes
3answers
69 views

Why is $\max(x, x')$ equivalent to $\frac{1}{2}( x + x' + |x - x' |)$?

Why is it that $$\max(x, x') = \frac{1}{2}( x + x' + |x - x'|)$$ is true? Is it supposed to be obvious? Because it seems to come out of thin air for me. Anyway, I've verified this by plotting it in ...
0
votes
1answer
27 views

Elementary number theory proofs using functions

The functions $f$ and $g$ are defined by $f(x) =$ remainder when $x^2$ is divided by $7$. $g(x) =$ remainder when $x^2$ is divided by $5$. (a) Show that $f(5)=g(3)$ (b) If $n$ is an integer, ...
1
vote
0answers
23 views

Periodicity of Newton's method approximations on a cubic polynomial

Bruckner & Bruckner, Elementary Real Analysis Let $f(x) = x^3 - 3x + 3$ Applying Newton's method to get $x_{n+1} = x_n -\frac{f(x)}{f'(x)} \ ,$ prove that for any positive integer $p$, there ...
-1
votes
1answer
21 views

Ratio known and Amount needed known

You have a a recipe for the perfect orange juice, it is: 26 parts Water to 1 Part Orange Juice (26:1). However overall you only want 20ml of mixed liquid (orange juice and water combined). What ...
3
votes
3answers
36 views

Finding periodic (trigonometric?) function given points

It's been a while since I've taken a math class. I need a couple functions for a program I'm working on. I can tell they involve trigonometry, but I can't figure out how to derive the function ...
0
votes
0answers
13 views

Function/algorithm to generate a random walk on a graph

I'm looking for a graph function or an algorithm that can generate a random fluctuating random walk that will eventually converge between the value of y = 0 and y = 1, more or less after a number of ...
1
vote
0answers
51 views

Is it possible to approximate $cos(x)$ with a linear combination of Gaussians $e^{-x^2}$?

I am interested in approximating $\cos x$ with a linear combination of $e^{-x^2}$. I am not an expert in approximation theory but there are a couple things that give me a bit of hope that it might be ...
0
votes
1answer
35 views

Confused by one-to-one question, I think it's order incorrectly

I have this question and it seems a tad redundant If $A$ and $B$ are infinite sets, is it possible for there to be a 1-1 function from $A$ to $B$ and a 1-1 function from $B$ to $A$ without there ...
0
votes
4answers
66 views

Why is this function a bijection?

Consider the function below $$f:\mathbb{R^+} \to \mathbb{R^+}$$ given by $$f(x) = \sqrt{x}$$. Now it makes sense that the function is injective because $f(x) = f(y) \implies \sqrt{x} = \sqrt{y} ...
1
vote
1answer
112 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
0
votes
1answer
63 views

Lazy mathematician: what are the real lengths in an Ideal Lambert quadrilateral?

At the moment it is to hot for real mathematics but I wanted to have a function that relates the lengths of the real sides of an Ideal Lambert quadrilateral An Ideal Lambert quadrilateral (my term, ...
0
votes
1answer
47 views

How to find the value of $2g(1)+2f(1)-h(1)$?

If $$\lim_{ m\to\infty }{ \frac { x^{ m }f(1)+h(x)+1 }{ 2x^m+3x+3 } }$$ is continuous at $x=1$ and $g(1)=\lim_{ x\to0}(\ln x)^{ 2/\ln(x) }$ then how to find the value of $2g(1)+2f(1)-h(1)$? Assume ...
0
votes
1answer
31 views

$f(x)=y$ while $g(y)=x$; Is it possible to find two not reverse functions that behave such at least for a given set of inputs and outputs?

I want to know if it is possible to program such a code that could determine two distinguish, not inverse, functions, say $f$ and $g$, that is true for the below statements at a given input and output ...
-11
votes
0answers
24 views

Please write down exact output after the following statements are proceesed [on hold]

Please write down exact output after the following statements are proceesed ...
0
votes
2answers
16 views

Invalid function or invalid domain

Let $ f : A \rightarrow B $ What happens if $\exists\ a\in A $ which doesn't map to any element in B ?
0
votes
0answers
8 views

Value distribution into random variables

I want to distribute a large number into quantities of 100s, 50s, 25s, 5s, 1s Lets say, I'm selling apples in quantities of 100s, 50s .... 1. Based on the quantity there is different pricing. 1 apple ...
0
votes
1answer
29 views

How to calculate powers of a permutation in cyclic notation? [on hold]

How do I calculate powers of an 8-cycle (1 2 3 4 5 6 7 8) ?
0
votes
2answers
10 views

Is the sum of a unimodal and increasing function still unimodal?

There is no specific function. I would like to know if there is information on summation of a unimodal and increasing function? (Continuous functions)
-2
votes
2answers
47 views

An injection from R × {0, 1} to R [on hold]

What would be an example of this An injection from R × {0, 1} to R i think it is all real numbers f(x) = x Can some one help me on this. Thanks in advance
1
vote
0answers
11 views

A property of Quasiconvex functions.

Let f be a strictly quasiconvex differentiable function and Df denote its gradient. Is the following implication true? :"Whenever f(y) < f(x), we also have (Df(x))'(y - x) < 0" . Suppose that f ...