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

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39
votes
5answers
5k views

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

How do I 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 ...
20
votes
3answers
964 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} ...
0
votes
1answer
487 views
+50

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 ...
3
votes
3answers
427 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 ...
8
votes
2answers
646 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 ...
1
vote
1answer
665 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 $a>0$ ...
18
votes
1answer
659 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
3answers
352 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 ...
89
votes
6answers
3k 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 ...
12
votes
4answers
2k 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?
13
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.
7
votes
4answers
437 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 ...
1
vote
2answers
151 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 ...
-5
votes
3answers
148 views

Give an example of a function $f: \mathbb{N} \rightarrow \mathbb{N}$ with the property that there exists [closed]

Give an example of a function $f: \mathbb{N} \rightarrow \mathbb{N}$ with the property that there exists a function $g: \mathbb{N} \rightarrow \mathbb{N}$ such that the composition $g \circ f$ is the ...
10
votes
3answers
13k 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 ...
5
votes
3answers
3k 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 ...
38
votes
4answers
2k 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 ...
20
votes
2answers
1k 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 $f(n+1)=f(n)+m$, ...
2
votes
3answers
481 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}: ...
2
votes
3answers
363 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 $f^{-1}(f(A))=A$?
12
votes
6answers
290 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?
5
votes
2answers
547 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 $f$ ...
4
votes
3answers
725 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?
26
votes
7answers
1k views

how to get $ f(x) $ if we know $ f(f(x))=x^2+x $

how to get $ f(x) $ if we know $ f(f(x))=x^2+x $ Is there elementary function of $f(x)$ satisfy the equation?
27
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
974 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 ...
1
vote
2answers
407 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 ...
6
votes
3answers
283 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 ...
4
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?
4
votes
7answers
1k 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
209 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
11k 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
131 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
104 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
3answers
700 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 = ...
7
votes
1answer
519 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
21
votes
7answers
1k views

Notation for repeated application of function

If I have the function $f(x)$ and I want to apply it $n$ times, what is the notation to use? For example, would $f(f(x))$ be $f_2(x)$, $f^2(x)$, or anything less cumbersome than $f(f(x))$? This is ...
6
votes
1answer
280 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 ...
3
votes
1answer
557 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 ...
12
votes
7answers
1k views

A nontrivial everywhere continuous function with uncountably many roots?

This is my first post on SE, forgive any blunders. I am looking for an example of a function $f:\mathbb{R} \to \mathbb{R}$ which is continuous everywhere but has uncountably many roots ($x$ such that ...
11
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 ...
5
votes
2answers
888 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 ...
8
votes
1answer
201 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 ...
5
votes
5answers
3k 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 ...
4
votes
3answers
1k views

Is there a systematic way of solving cubic equations?

In my text book, to solve cubic equations, I need to find by trial & error what $f(a)$ will make the equation 0. The factor will be $(x-a)$ then the other factor will be $Ax^2+Bx+C$ then I can ...
3
votes
0answers
754 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
4answers
339 views

Why $f^{-1}(f(A)) \not= A$ [duplicate]

Let $A$ be a subset of the domain of a function $f$. Why $f^{-1}(f(A)) \not= A$. I was not able to find a function $f$ which satisfies the above equation. Can you give an example or hint. I was asking ...
4
votes
2answers
1k views

Bijection from finite (closed) segment of real line to whole real line

Is there a bijection from a finite (closed) segment of the real line to $\mathbb{R}$? For example, is there a bijection from $[0,1]$ to $\Bbb{R}$? If so, is there a straightforward example? If not, ...
4
votes
6answers
730 views

Writing a function $f$ when $x$ and $f(x)$ are known

I'm trying to write a function. For each possible input, I know what I want for output. The domain of possible inputs is small: $$\begin{vmatrix} x &f(x)\\ 0 & 2\\ 1 ...
2
votes
2answers
775 views

Parallel functions.

In 2 dimensions, we can draw 2 parallel lines that have the same distance from a line. I wanted to find parallel functions of a function and their distance is $d$ to the function for all inputs and ...