Questions tagged [functions]

For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.

Filter by
Sorted by
Tagged with
182
votes
8answers
51k 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 that'...
158
votes
1answer
3k views

What functions can be made continuous by "mixing up their domain"?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ such that $f\circ \phi$ is continuous. So one could say a potentially ...
154
votes
4answers
52k 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?
139
votes
1answer
5k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where $...
68
votes
5answers
12k views

Why do engineers use derivatives in discontinuous functions? Is it correct?

I am a Software Engineering student and this year I learned about how CPUs work, it turns out that electronic engineers and I also see it a lot in my field, we do use derivatives with discontinuous ...
68
votes
6answers
5k views

How to straighten a parabola?

Consider the function $f(x)=a_0x^2$ for some $a_0\in \mathbb{R}^+$. Take $x_0\in\mathbb{R}^+$ so that the arc length $L$ between $(0,0)$ and $(x_0,f(x_0))$ is fixed. Given a different arbitrary $a_1$, ...
66
votes
18answers
8k views

Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$ [closed]

I'm thinking about this question in the sense that we often have a term $(-1)^n$ for an integer $n$, so that we get a sequence $1,-1,1,-1...$ but I'm trying to find an expression that only gives every ...
64
votes
5answers
14k 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 ...
63
votes
6answers
23k views

Functions that are their own inverse.

What are the functions that are their own inverse? (thus functions where $ f(f(x)) = x $ for a large domain) I always thought there were only 4: $f(x) = x , f(x) = -x , f(x) = \frac {1}{x} $ and $ ...
61
votes
6answers
6k views

Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$...

I've been stumped by this problem: Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that $$f\circ g=h$$ $$g\circ h=f$$ $$h\circ f=g$$ or prove that no ...
61
votes
8answers
94k views

How do I prove that a function is well defined?

How do you in general prove that a function is well-defined? $$f:X\to Y:x\mapsto f(x)$$ I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
58
votes
4answers
4k views

Can there be an injective function whose derivative is equivalent to its inverse function?

Let's say $f:D\to R$ is an injective function on some domain where it is also differentiable. For a real function, i.e. $D\subset\mathbb R, R\subset\mathbb R$, is it possible that $f'(x)\equiv f^{-1}(...
58
votes
6answers
8k views

Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx $ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
57
votes
6answers
10k views

Big O Notation "is element of" or "is equal"

People are always having trouble with "big $O$" notation when it comes to how to write it down in a mathematically correct way. Example: you have two functions $n\mapsto f(n) = n^3$ and $n\mapsto g(n)...
56
votes
6answers
13k views

Do harmonic numbers have a “closed-form” expression?

One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
56
votes
11answers
28k views

What is an operator in mathematics?

Could someone please explain the mathematical difference between an operator (not in the programming sense) and a function? Is an operator a function?
54
votes
9answers
7k views

On the functional square root of $x^2+1$

There are some math quizzes like: find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$ such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$ If such $\phi$ exists (it does in this example), $\phi$ can ...
53
votes
6answers
47k views

Create unique number from 2 numbers

is there some way to create unique number from 2 positive integer numbers? Result must be unique even for these pairs: 2 and 30, 1 and 15, 4 and 60. In general, if I take 2 random numbers result must ...
52
votes
3answers
1k views

What numbers can be created by $1-x^2$ and $x/2$?

Suppose I have two functions $$f(x)=1-x^2$$ $$g(x)=\frac{x}{2}$$ and the number $1$. If I am allowed to compose these functions as many times as I like and in any order, what numbers can I get to if I ...
50
votes
4answers
5k 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} \...
50
votes
6answers
3k views

Proving that a function is odd

Assume that there exists a function $f:\mathbb{R}\to\mathbb{R}$ that is bijective and satisfies $$ f(x) + f^{-1}(x)=x $$ for all $x$. Here $f^{-1}$ is the inverse function. Show that $f$ is odd. This ...
49
votes
8answers
28k views

When to Stop Using L'Hôpital's Rule

I don't understand something about L'Hôpital's rule. In this case: $$ \begin{align} & {{}\phantom{=}}\lim_{x\to0}\frac{e^x-1-x^2}{x^4+x^3+x^2} \\[8pt] & =\lim_{x\to0}\frac{(e^x-1-x^2)'}{(x^4+...
49
votes
6answers
46k views

Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist

I've been trying to solve the following problem: Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\...
48
votes
7answers
5k views

A function in which addition and multiplication behave the same way

Exponents have a well-known property: $$x^ax^b = x^{a+b}$$ but $$x^{a} + x^{b} \neq x^{a+b}$$ Similarly, $$\log(a) + \log(b) = \log(ab) $$ But $$\log(a)\log(b) \neq \log(ab)$$ So my question ...
48
votes
3answers
8k views

Is it possible for the derivative of a function to grow arbitrarily faster than the function itself?

We know that there exist some functions $f(x)$ such that their derivative $f'(x)$ is strictly greater than the function itself. for example the function $5^x$ has a derivative $5^x\ln(5)$ which is ...
48
votes
2answers
1k views

Looking for a function such that...

There was this question on one of the whiteboards at my company, and I found it intriguing. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Anyway, here it is: ...
48
votes
1answer
1k views

What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
47
votes
4answers
1k views

$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor$ is true?

I found the following relational expression by using computer: For any natural number $n$, $$\lfloor \sqrt n+\sqrt {n+1}+\sqrt{n+2}+\sqrt{n+3}+\sqrt{n+4}\rfloor=\lfloor\sqrt {25n+49}\rfloor.$$ Note ...
46
votes
10answers
2k views

Dog bone-shaped curve: $|x|^x=|y|^y$

EDITED: Some of the questions are ansered, some aren't. EDITED: In order not to make this post too long, I posted another post which consists of more questions. Let $f$ be (almost) the implicit curve$...
45
votes
11answers
20k views

What is a function?

I have been quite confused by the definition of functions and their uses.. First of all can one define functions in a clear understandable way, with a clear explanation of their uses, how they work ...
45
votes
4answers
21k views

Is composition of measurable functions measurable?

We know that if $ f: E \to \mathbb{R} $ is a Lebesgue-measurable function and $ g: \mathbb{R} \to \mathbb{R} $ is a continuous function, then $ g \circ f $ is Lebesgue-measurable. Can one replace the ...
44
votes
11answers
7k views

How can a "proper" function have a vertical slope?

Plotting the function $f(x)=x^{1/3}$ defined for any real number $x$ gives us: Since $f$ is a function, for any given $x$ value it maps to a single y value (and not more than one $y$ value, because ...
44
votes
3answers
10k views

What is the "fastest" increasing function that's useful in some area of math?

Context: I just completed the first quarter of an Intro to Real Analysis class, and while I was thinking about how some functions (like $x^2$) aren't uniformly continuous because they, roughly ...
44
votes
3answers
16k views

What is a basis for the vector space of continuous functions?

A natural vector space is the set of continuous functions on $\mathbb{R}$. Is there a nice basis for this vector space? Or is this one of those situations where we're guaranteed a basis by invoking ...
44
votes
2answers
760 views

Which functions satisfy $f^n(x) = f(x)^n$ for some $n \ge 2$?

Let $n$ be an integer greater than $1$. The notation $f^n$ is notoriously ambiguous: it means either the $n$-th iterate of $f$ or its $n$-th power. I was wondering when the two interpretations are in ...
43
votes
7answers
4k 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?
43
votes
7answers
55k views

What does it mean when two functions are "orthogonal", why is it important?

I have often come across the concept of orthogonality and orthogonal functions e.g in fourier series the basis functions are cos and sine, and they are orthogonal. For vectors being orthogonal means ...
42
votes
7answers
6k views

Why is $\log(\sqrt{x^2+1}+x)$ odd?

$$f(x) = \log(\sqrt{x^2+1}+x)$$ I can't figure out, why this function is odd. I mean, of course, its graph shows, it's odd, but when I investigated $f(-x)$, I couldn't find way to $-\log(\sqrt{x^2+1}+...
42
votes
6answers
18k views

Why is there no function with a nonempty domain and an empty range?

Let $A$ to be a nonempty set and $B= \emptyset$; then $ A \times B$ is a set. And let $F$ be a function $A \to B$. Then $F \subseteq A \times B$. By the axiom of specification, $F$ must exists (if I ...
41
votes
3answers
19k views

Why is an empty function considered a function?

A function by definition is a set of ordered pairs, and also according the Kuratowski, an ordered pair $(x,y)$ is defined to be $$\{\{x\}, \{x,y\}\}.$$ Given $A\neq \varnothing$, and $\varnothing\...
40
votes
13answers
13k views

Function which creates the sequence 1, 2, 3, 1, 2, 3, ...

I was wondering how to map the set $\mathbb{Z}^+$ to the sequence $1, 2, 3, 1, 2, 3, \ldots$. I thought it would be easy, but I was only able to obtain an answer through trial and error. For a ...
40
votes
11answers
10k views

How is $e^x$ read aloud?

My current research colleague from New Castle told me that I was reading it wrong. I usually read it as e power x. How do you read aloud $e ^ x$? Is it: e raised to x e power x e powered x or e ...
40
votes
4answers
45k views

What is the difference between $\arg\max$ and $\max$?

What is the exact difference between $\arg\max$ and $\max$ of a function? Is it right to say the following? $\arg\max f(x)$ is nothing but the value of $x$ for which the value of the function is ...
40
votes
5answers
470k views

How to determine if a function is one-to-one?

I am looking for the "best" way to determine whether a function is one-to-one, either algebraically or with calculus. I know a common, yet arguably unreliable method for determining this answer would ...
40
votes
4answers
58k 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 ...
40
votes
7answers
15k views

Are there other kinds of bump functions than $e^\frac1{x^2-1}$?

I've only seen the bump function $e^\frac1{x^2-1}$ so far. Where could I find examples of functions $C^∞$ on $\mathbb{R}$ that are zero everywhere except on $(-1,1)$? Are there others that do not ...
39
votes
6answers
18k views

Is there a name for the function $\max(x, 0)$?

Is there a name for the function $ \max(x, 0) $? For comparison, the function $ \max(x, -x) $ is known as the absolute value (or modulus) of x, and has its own notation $ |x| $
39
votes
3answers
5k 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 $f(n+1)=f(n)+m$, ...
39
votes
11answers
22k 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 ...
39
votes
7answers
1k 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?

1
2 3 4 5
590