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

learn more… | top users | synonyms (1)

8
votes
2answers
96 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
8
votes
1answer
410 views

To get addition formula of $\tan (x)$ via analytic methods

Assume that we only know $\tan (0)=0$ and also given the relation $\tan'(x)=1+\tan^2(x)$ about $\tan (x)$ and we do not know other $\tan (x)$ relations of trigonometry. How can I get the additon ...
8
votes
1answer
90 views

Are there associative bijections $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$?

This is my first question and I hope this question is not too brief to be acceptable: There are well-known bijections $\mathbb{N \times N \to N}$ and well-known associative compositions on countably ...
8
votes
2answers
412 views

if $|f(n+1)-f(n)|\leq 2001$, $|g(n+1)-g(n)|\leq 2001$, $|(fg)(n+1)-(fg)(n)|\leq 2001$ then $\min\{f(n),g(n)\}$ is bounded

The following question was proposed at MOP 2001 A function $f:\mathbb{N}\to\mathbb{N}$ is called cautious if $|f(n+1)-f(n)|\leq 2001$ for all $ n\in\mathbb{N} $. Suppose that $f,g,h$ are ...
8
votes
2answers
11k views

Proving a function is onto and one to one

I'm reading up on how to prove if a function (represented by a formula) is one-to-one or onto, and I'm having some trouble understanding. To prove if a function is one-to-one, it says that I have to ...
8
votes
3answers
181 views

How do I read this question? (subject: bijections)

Introduction In Basic Algebra I, I am struggling with fully understanding the following exercise: Show that $S\overset{\alpha}{\to}T$ is injective if and only if there is a map ...
8
votes
1answer
235 views

If the set of values , for which a function has positive derivative , is dense then is the function increasing?

Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $A:=${ $x \in \mathbb R :f'(x)>0$ } is dense in $\mathbb R$ , then is it true that $f$ is an increasing function ? What ...
8
votes
1answer
135 views

If the function $f$ satisfies the equation $f(xf(y)+x)=xy+f(x)$, find $f$

Question Let the function $f:\mathbb R\to\mathbb R$,and such $$f(xf(y)+x)=xy+f(x)$$ Find all $f(x)$ Let $x=1,y=1$,then $$f(f(1)+1)=1+f(1)$$ let $f(1)=t$,then $$f(t+1)=1+t$$ So I guess ...
8
votes
3answers
3k views

Change of order of double limit of function sequence

The more general quesion is under what conditions the folloing equality will hold (all functions are real valued): $$\lim_{x \rightarrow a} \ \lim_{j \rightarrow \infty} f_j(x) = \lim_{j \rightarrow ...
8
votes
2answers
306 views

$ \sum\limits_{i=1}^{p-1} \Bigl( \Bigl\lfloor{\frac{2i^{2}}{p}\Bigr\rfloor}-2\Bigl\lfloor{\frac{i^{2}}{p}\Bigr\rfloor}\Bigr)= \frac{p-1}{2}$

I was working out some problems. This is giving me trouble. If $p$ is a prime number of the form $4n+1$ then how do i show that: $$ \sum\limits_{i=1}^{p-1} \Biggl( ...
8
votes
1answer
141 views

If $f:X \to X$ is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous?

If $f:X \to X$ (codomain and domain have the same topology) is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous? Note that the orbit being finite and $f$ being a ...
8
votes
1answer
238 views

Prove that : $|f(b)-f(a)|\geqslant (b-a) \sqrt{f'(a) f'(b)}$ with $(a,b) \in \mathbb{R}^{2}$

Let $(a,b) \in \mathbb{R}^{2}$ such that $a<b$ and $f\in C^2([a,b],\mathbb{R})$ such that $f'\neq 0$ and $f''/f'$ is decreasing. Prove that : $$|f(b)-f(a)|\geqslant (b-a) ...
8
votes
1answer
73 views

Is every real valued function on an interval a sum of two functions with Intermediate Value Property?

If $I$ is an interval of real numbers , then is it true that any function $f:I \to \mathbb R$ can be written as $f=f_1+f_2$ , where $f_1 , f_2 : I \to \mathbb R$ have the Intermediate value property? ...
8
votes
1answer
97 views

A question of rationality

This problem was asked to me by a friend and I simply have no idea about it. So I have not progressed a single bit. The problem is this: If $f :\mathbb{R}\to \mathbb{R}$ is an infinitely ...
8
votes
4answers
260 views

Why isn't $y=(x^6)^{1/3}$ a polynomial function?

I've been told that $y=(x^6)^{1/3}$ isn't a polynomial function because of the radical but I believe that the equation could be simplified to $y=x^2$ which fits the definition of a polynomial ...
8
votes
1answer
211 views

Function mapping challange

For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$ and $f(k) = f(f(k + 1))$. This is ...
8
votes
2answers
351 views

Prove the following property of $f(x)$?

Let $$f(x)=|a_1\sin(x)+a_2\sin(2x)+a_3\sin(3x)+...+a_n\sin(nx)|.$$ Given that $f(x)$ is less than or equal to $|\sin(x)|$ for all $x$, prove that $|a_1+a_2+a_3+....|$ is less than or equal to ...
8
votes
1answer
94 views

How many expressions can be formed with two commutative and associative functions?

EDIT: I have posted a generalization of this question to MathOverflow here. Suppose we have two binary functions $f,g$ which are commutative and associative, i.e., satisfying $$ f(a,b) = f(b,a) ...
8
votes
1answer
55 views

About continuity

one question is disturbing me : let f and g two continuous (real-valued) functions on the unit interval [0,1] with the property that $[f(x)-f(y)][g(x)-g(y)]=0 ,\forall x,y \in [0,1]$. To my ...
8
votes
0answers
236 views

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
8
votes
0answers
341 views

Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
8
votes
0answers
344 views

Multiplicative Identity analog for absolute value

Is there a standard name for the function: $$ f(x) = \begin{cases} x & \text{if |x|≤1;}\\ 1/x & \text{if |x|>1;}\\ \end{cases} $$ And is there a potential application of this ...
8
votes
2answers
199 views

How can we prove this function must be linear?

Let $f:[a,b]\to\mathbb{R}$ be continuous. Suppose for any sequence $(r_n)_{n=0}^{\infty}$ with $\lim_{n\to\infty}r_n=0$, and any $x\in(a,b)$: ...
7
votes
6answers
964 views

Do we have always $f(A \cap B) = f(A) \cap f(B)$?

Suppose $A$ and $B$ are subsets of a topological space and $f$ is any function from $X$ to another topological space $Y$. Do we have always $f(A \cap B) = f(A) \cap f(B)$? Thanks in advance
7
votes
6answers
3k views

Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective

Can anyone come up with an explicit example of two functions $f$ and $g$ such that: $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective? I tried the following: $$f:\mathbb{R}\rightarrow ...
7
votes
6answers
4k views

Max Min of function less than Min max of function

I can't understand why max min of a function is less than equal to min max of that function i.e Why $$\underset{x}{\text{max}}\:\underset{y}{\text{min}} f(x,y) \leq ...
7
votes
6answers
1k views

What do I not understand about one-to-one functions?

Firstly, a definition: Definition 1: A function $\phi : X \rightarrow Y$ is one-to-one if $\phi(x_1) = \phi(x_2)$ only when $x_1 = x_2$. Now the question: Students often misunderstand the ...
7
votes
7answers
378 views

Function satisfying $x = f(f(x))$ and $x \not= f(x)$

Is there a function that would satisfy the following conditions?: $\forall x \in X, x = f(f(x))$ and $x \not= f(x)$, where the set $X$ is the set of all triplets $(x_1,x_2,x_3)$ with $x_i \in ...
7
votes
3answers
499 views

If $f \circ g = f$, prove that $f$ is a constant function.

Suppose $A$ is a nonempty set and $f: A \rightarrow A$ and for all $g:A \rightarrow A,$ $f \circ g = f$. Prove that $f$ is a constant function. This result seems obvious, but I can't seem to find ...
7
votes
2answers
731 views

Intuition behind arc length formula

I understand the arc length formula is derived from adding the distances between a series of points on the curve, and using the mean value theorem to get: $ L = \int_a^b \sqrt{ 1 + (f'(x))^2 } dx ...
7
votes
2answers
326 views

Must $g$ be the identity if $f = g \circ f$?

I am finding it hard to solve the following problem. Let $A$ is a set and $f : A \rightarrow A$ and $g : A \rightarrow A$. If $f = g \circ f$, must $g$ be an identity function always? Will there be ...
7
votes
9answers
420 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...
7
votes
6answers
217 views

Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$

Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function. Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the ...
7
votes
2answers
243 views

Does $F(t)F'(t) \le 0$ imply that $F$ does not change signs?

The question is in the title, really. More precisely, suppose $F:[0,1] \to \mathbb{R}$ is continuously differentiable and satisfies (i) $F(t)F'(t) \le 0$ for all $t \in [0,1]$ (ii) $F(0) = 1$ ...
7
votes
4answers
666 views

Solving the functional equation $f(x+1) - f(x-1) = g(x)$

Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies $$ f(x+1) - f(x-1) = g(x) $$
7
votes
5answers
553 views

Functional Notation.

I have some doubts regarding function notation: First If I present a function I write:$f(x)$ If I write it's inverse:$f^{-1}(x)$ So why doesn't$f(f(x))=f^2(x)$ Second If $\frac{df(x)}{dx}=f'(x)$ ...
7
votes
5answers
255 views

Expressing in terms of symmetric polynomials.

How to express $$a^7+b^7+c^7$$ in terms of symmetric polynomials ${\sigma}_{1}=a+b+c$, ${\sigma}_{2}=ab+bc+ca$ and ${\sigma}_{3}=abc$ ?
7
votes
3answers
181 views

Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?

True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
7
votes
2answers
331 views

A differentiation question conceptual query

I'm quite unsure about how to deal with differentiation of absolute functions, and their continuity. For example, the question I was dealing with was the following: $$ f(x) = \frac{x}{1 + |x|}$$ ...
7
votes
2answers
267 views

Is there a non-constant smooth function mapping $\mathbb{R}$ into $\mathbb{Q}$

I cannot think of a non-constant smooth function which maps all real numbers into rational numbers. Can anyone give a simple example ? The simpler, the better !
7
votes
6answers
441 views

Rigorous Definition of “Function of”

When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a ...
7
votes
2answers
10k views

Number of onto functions

What are the number of onto functions from a set $\Bbb A $ containing m elements to a set $\Bbb B$ containing n elements. I ...
7
votes
2answers
4k views

Ease-in-out function

I am trying to create a nice ease-in-out function that given values from 0 - 1 produces an output of 0 - 1 which accelerates slowly up to full speed then slows down again as it nears 1. I currently ...
7
votes
6answers
333 views

Removable discontinuity question

A quick and easy question on terminology: If we have a function $$f(x) = \frac{x(x-2)}{(x-2)}$$ then clearly the function is not defined at $x = 2$. If we cancel out "$(x-2)$" then the function is ...
7
votes
3answers
385 views

What's the difference between a bijection and an isomorphism? [duplicate]

I don't understand what the difference is between a bijection and an isomorphism. They seem to both just be a invertible mapping. Is the set of all bijections a subset of isomorphisms? Or vice ...
7
votes
3answers
394 views

If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
7
votes
2answers
96 views

$f: \mathbb N \to \mathbb N $ is injective $ \implies $ $\lim\Big(f(n)\Big)=+ \infty $?

If $f: \mathbb N \to \mathbb N $ is an injective function , then for the sequence $\Big(f(n)\Big)$ do we have $\lim\Big(f(n)\Big)=+ \infty $ ?
7
votes
1answer
316 views

Arithmetic function to return lowest in-parameter

Is there a mathematical function such that; f(3, 5) = 3 f(10, 2) = 2 f(14, 15) = 14 f(9, 9) = 9 It would be even more cool if there's a function that takes ...
7
votes
1answer
363 views

Can functions have multiple inputs?

Now bear with me here, I'm not the best at math. I'm just trying to find out something that I never really learned. I was wondering, can a function have multiple inputs such as this one below? ...
7
votes
1answer
657 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.