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

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46
votes
5answers
7k views

How to define a bijection between $(0,1)$ and $(0,1]$?

How to define a bijection between $(0,1)$ and $(0,1]$? Or any other open and closed intervals? If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if ...
22
votes
3answers
1k views

No continuous function switches $\mathbb{Q}$ and the irrationals

Is there a way to prove the following result using connectedness? Result: Let $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} ...
3
votes
3answers
547 views

Prove $f(S \cup T) = f(S) \cup f(T)$

$f(S \cup T) = f(S) \cup f(T)$ f(S) encompasses all x that is in S f(T) encompasses all x that is in T thus the domain being the same, both the LHS and RHS map to the same ys, since the function ...
12
votes
4answers
4k views

Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
0
votes
1answer
741 views

Integers and integer functions

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
11
votes
3answers
16k views

How do I divide a function into even and odd sections?

While working on a proof showing that all functions limited to the domain of real numbers can be expressed as a sum of their odd and even components, I stumbled into a troublesome roadblock; namely, I ...
8
votes
2answers
747 views

On sort-of-linear functions

Background A function $ f: \mathbb{R}^n \rightarrow \mathbb{R} \ $ is linear if it satisfies $$ (1)\;\; f(x+y) = f(x) + f(y) \ , \ and $$ $$ (2)\;\; f(\alpha x) = \alpha f(x) $$ for all $ x,y \in ...
6
votes
3answers
4k views

Injective and Surjective Functions

Let $f$ and $g$ be functions such that $f\colon A\to B$ and $g\colon B\to C$. Prove or disprove the following a) If $g\circ f$ is injective, then $g$ is injective Here's my proof that this ...
4
votes
3answers
1k views

Surjectivity of Composition of Surjective Functions

Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto?
1
vote
1answer
758 views

continuous functions on $\mathbb R$ such that $g(x+y)=g(x)g(y)$

Let $g$ be a function on $\mathbb R$ to $\mathbb R$ which is not identically zero and which satisfies the equation $g(x+y)=g(x)g(y)$ for $x$,$y$ in $\mathbb R$. $g(0)=1$. If $a=g(1)$,then $a>0$ ...
19
votes
1answer
722 views

Characterising functions $f$ that can be written as $f = g \circ g$?

I'd like to characterise the functions that ‘have square roots’ in the function composition sense. That is, can a given function $f$ be written as $f = g \circ g$ (where $\circ$ is function ...
4
votes
4answers
434 views

Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality

Let $f: A\longrightarrow B$ be a function. 1)Prove that for any two sets, $C,D\subseteq A$ , we have $f(C) \setminus f(D)\subseteq f(C\setminus D)$. 2)Give an example of a function $f$, and sets ...
10
votes
5answers
7k views

Calculating the total number of surjective functions

It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ...
96
votes
7answers
4k views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?

I've been perusing the internet looking for interesting problems to solve. I found the following problem and have been going at it for the past 30 minutes with no success: Find a continuous ...
23
votes
2answers
2k views

Is there a natural way to extend repeated exponentiation beyond integers?

This question has been in my mind since high school. We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, ...
15
votes
6answers
1k views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
39
votes
4answers
3k views

Nice expression for minimum of three variables?

As we saw here, the minimum of two quantities can be written using elementary functions and the absolute value function. $\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$ There's even a nice intuitive ...
8
votes
4answers
682 views

Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$?

Is this proof correct? To prove $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functions $F$. Let any number $y\in F(A)\cap F(B)$. We want to show $y\in F(A\cap B).$ Therefore, $y\in F(A)$ and ...
14
votes
6answers
357 views

Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist?
6
votes
2answers
862 views

Right Inverse for Surjective Function

Prove that if $f:X\to Y$ is a surjective function between sets, then there must exist a function $g:Y\rightarrow X$ such that $f\circ g=1_Y$. I know that the identity function is onto, and if $f$ ...
2
votes
2answers
836 views

Proof for the formula of sum of arcsine functions $ \arcsin x + \arcsin y $

It is known that the following holds good: $$ \arcsin x + \arcsin y \\ \begin{align} &=\arcsin( x\sqrt{1-y^2} + y\sqrt{1-x^2}) \;\;;x^2+y^2 \le 1 \;\text{ or }\; x^2+y^2 > 1, xy< 0\\ ...
1
vote
2answers
200 views

Let $(a,b)$ and $(c,d)$ be intervals in $\Bbb R$, and find an injective and surjective function from $(a,b)$ to $(c,d)$

So here is this question I got stuck on: Let $(a,b)$, $(c,d)$ be intervals (not sure if that's the correct term) on $\Bbb R$, so that $a<b$, $c<d$. Find an injective and surjective function ...
6
votes
3answers
320 views

bijection between $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ [duplicate]

I understand that both $\mathbb{N}$ and $\mathbb{N}\times\mathbb{N}$ are of the same cardinality by the Shroeder-Bernstein theorem, meaning there exists at least one bijection between them. But I ...
5
votes
2answers
2k views

Surjectivity implies injectivity

Let S be a finite set.Let F be a surjective function from S to S. How do I prove that it is injective?
30
votes
7answers
1k views

How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?

How to get $f(x)$, if we know that $f(f(x))=x^2+x$? Is there an elementary function $f(x)$ that satisfies the equation?
5
votes
1answer
1k views

Construct a monotone function which has countably many discontinuities

I read in a textbook, which had seemed to have other dubious errors, that one may construct a monotone function with discontinuities at every point in a countable set $C \subset [a,b]$ by enumerating ...
5
votes
1answer
975 views

If $|A|=30$ and $|B|=20$, find the number of surjective functions $f:A \to B$.

Let there be: $|A|=n$ and $|B|=m$ if $m>n$ then there are $$m(m-1)\cdots(m-n+1)$$ injective functions, so in this case we have $|A|=30$ and $|B|=20$ that means $m<n$ so there exists a ...
2
votes
3answers
430 views

Is $f^{-1}(f(A))=A$ always true?

If we have a function $f:X\rightarrow Y$ where $A\subset X$, is it true to say that $f^{-1}(f(A))=A$?
6
votes
2answers
995 views

Is there a Cantor-Schroder-Bernstein statement about surjective maps?

Let $A,B$ be two sets. The Cantor-Schroder-Bernstein states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$. I was ...
2
votes
3answers
734 views

Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$

I am struggling to prove this map statement on sets. The statement is: Let $f:X \rightarrow Y$ be a map. i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$ ii) $\forall_{A,B \subset X}: ...
10
votes
2answers
651 views

Why $f(x) = \sqrt{x}$ is a function?

Why $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is a ...
5
votes
5answers
4k views

How to prove $(f \circ\ g) ^{-1} = g^{-1} \circ\ f^{-1}$?

I'm doing exercise on discrete mathematics and I'm stuck with question: If $f:Y\to Z$ is an invertible function, and $g:X\to Y$ is an invertible function, then the inverse of the composition $(f ...
3
votes
0answers
1k views

Left inverse iff injective; right inverse iff surjective

For a function $f:A\to B$, the function $g:B\to A$ is called: a left inverse for $f$ if $g\circ f$ is the identity on $A$ (i.e., $g\circ f = {\rm id}_A$); and a right inverse for $f$ if ...
4
votes
3answers
4k views

Why Is a Function Defined As Having Only One Y-Value Output?

I know that it is defined such that there is no more than one y-value output for any given x-value input, but I"m wondering WHY it is defined that way? Why can't we apply everything we know about ...
29
votes
11answers
4k views

How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
8
votes
3answers
1k views

Proof of a simple property of real, constant functions.

I recently came across the following theorem: $$ \forall x_1, x_2 \in \mathbb{R},\textrm{function, } f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto y; \ |f(x_1) - f(x_2)| \leq (x_1-x_2)^2 \implies ...
8
votes
2answers
365 views

Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$

Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that $|f'(x)| \le A/2$ for every $x \in [0,1]$. ...
8
votes
4answers
891 views

Proof of linear independence of $e^{at}$

Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$, form a linearly independent set ...
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 ...
1
vote
2answers
423 views

Expanded concept of elementary function?

After searching about why $\int e^{x^2}$ is not an elementary function, I was disappointed that I should understand about Galois theory, but then I started to think about a concept that treats ...
12
votes
6answers
1k views

In written mathematics, is $f(x)$a function or a number?

I often see notation/wording like "let $f(x)$ be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$". I would say that $\sin$ and $x \mapsto \sin(x)$ are functions, while $\sin(x)$ is a real ...
11
votes
6answers
637 views

How is the Integral of $\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$

Can Some one tell me what this method is called and how it works With a detailed proof $$\int_a^bf(x)~dx=\int_a^bf(a+b-x)~dx$$ I've been using this a lot in definite integration but haven't seemed ...
4
votes
7answers
2k views

How do I come up with a function to count a pyramid of apples?

My algebra book has a quick practical example at the beginning of the chapter on polynomials and their functions. Unfortunately it just says "this is why polynomial functions are important" and moves ...
2
votes
2answers
244 views

What are the strategies I can use to prove $f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$?

$f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$ I think I have to show that the LHS is a subset of the RHS and the RHS is a subset of the LHS, but I don't know how to do this exactly.
2
votes
3answers
16k views

If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one (given that…) [duplicate]

Possible Duplicate: Injective and Surjective Functions If g(f(x)) is one-to-one (injective) show f(x) is also one-to-one given that $f$ is a function from A to B and $g$ a function from B ...
0
votes
3answers
153 views

Finding the range of $f(x) = 1/((x-1)(x-2))$

I want to find the range of the following function $$f(x) = \frac{1}{(x-1)(x-2)} $$ Is there any way to find the range of the above function? I have found one idea. But that is too critical. Please ...
3
votes
2answers
110 views

To prove : If $f^n$ has a unique fixed point $b$ then $f(b)=b$

If $f: \mathbb R \to \mathbb R$ be a function such that for some $n_o \in \mathbb N$ , the $n_o$th iterate of $f$ has a unique fixed point $b$ , then how to prove that $f(b)=b$ ? I cant think of ...
3
votes
2answers
107 views

Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $

Let $f$ a continous function defined in the interval $[0,1]$. Assuming: $\forall x \in [0,1]:f(x) > x$ Prove: $\forall x \in [0,1]:f(x) > x + \varepsilon $ I tried to use Heine–Cantor theorem ...
2
votes
1answer
39 views

Constructing a quasiconvex function

Let $C\subset\mathbb{R}^2$ be a nonempty convex set. A function $f:C\rightarrow\mathbb{R}$ is called convex if $$ f(\lambda u+(1-\lambda)v)\leq\lambda f(u)+(1-\lambda)f(v), \quad\forall u,v\in C, ...
2
votes
3answers
880 views

Define a graph with segments or boundaries

This question is quite simple relative to the normal content on this site. (and please rename the title as I don't know what the proper definition is) Is it possible to have a function such as: $y = ...