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

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36
votes
5answers
4k 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 ...
19
votes
3answers
904 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
351 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 ...
1
vote
1answer
628 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 $a>0$ ...
17
votes
1answer
646 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 ...
83
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 ...
8
votes
2answers
600 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 ...
-5
votes
3answers
134 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 ...
4
votes
3answers
287 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 ...
7
votes
4answers
319 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 ...
9
votes
3answers
11k 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
2k 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 ...
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 $f(n+1)=f(n)+m$, ...
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 ...
12
votes
6answers
898 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.
3
votes
3answers
464 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
2answers
118 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 ...
26
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 ...
10
votes
4answers
1k 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?
6
votes
3answers
269 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 ...
2
votes
3answers
364 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}: ...
4
votes
7answers
862 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
181 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
9k 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
109 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
98 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 ...
8
votes
3answers
912 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 ...
21
votes
7answers
945 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 ...
7
votes
1answer
468 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
6
votes
1answer
264 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
423 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 ...
1
vote
2answers
393 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
7answers
947 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
996 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
834 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
200 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 ...
4
votes
2answers
1k 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
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 ...
1
vote
3answers
324 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 $f^{-1}(f(A))=A$?
4
votes
2answers
837 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
668 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
692 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 ...
2
votes
3answers
615 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 = ...
1
vote
3answers
326 views

Fixed point in a continuous function

Suppose that $f$ is a function defined in $[a;b]$ to $[a;b]$ and continuous on $[a;b]$. The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that ...
17
votes
3answers
664 views

What kind of “mathematical object” are limits?

When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly ...
9
votes
2answers
3k views

Constructing Continuous functions at given points

Ok. This question may sound very easy, but actually i am in great need of it. I have been facing trouble in constructing functions, which are only continuous at some particular sets. For e.g, the ...
4
votes
2answers
3k views

Are the smooth functions dense in either $\mathcal L_2$ or $\mathcal L_1$?

Is the subset consisting of all integrable (or square integrable) smooth functions of the set of all integrable (or square integrable) functions, dense under the usual Euclidean or integral of ...
16
votes
7answers
5k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...
13
votes
3answers
467 views

Is there any “superlogarithm” or something to solve $x^x$? [duplicate]

Is there any "superlogarithm" or something to solve an equation like this: $$x^x = 10?$$
11
votes
5answers
1k views

Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?

(source for above graph) (source for above graph) Both functions simplify to x, but why aren't the graphs the same?