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

learn more… | top users | synonyms (1)

114
votes
1answer
4k 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 ...
104
votes
8answers
5k 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 ...
79
votes
5answers
13k 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 ...
50
votes
6answers
5k 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 ...
47
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 ...
46
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 ...
43
votes
5answers
3k views

Functions that are their own inversion. [duplicate]

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 $ ...
43
votes
2answers
798 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: ...
42
votes
4answers
4k 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 ...
37
votes
8answers
9k 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] & ...
36
votes
11answers
4k 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 ...
36
votes
4answers
809 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 ...
35
votes
7answers
3k 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 ...
35
votes
4answers
4k 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 ...
34
votes
8answers
2k 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?
34
votes
5answers
1k views

When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$?

The question is: When $f(x+1)-f(x)=f'(x)$, what are the solutions for $f(x)$? The most obvious solution is a linear function of the form $f(x)=ax+b$. Is this the only solution? Edit I should ...
34
votes
3answers
3k 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 ...
32
votes
4answers
2k views

Why are removable discontinuities even discontinuities at all?

If I have, for example, the function $$f(x)=\frac{x^2+x-6}{x-2}$$ there will be a removable discontinuity at $x=2$, yes? Why does this discontinuity exist at all if the function can be simplified to ...
32
votes
6answers
1k views

What is the derivative of: $f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$?

I happened to ponder about the differentiation of the following function: $$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$ Now, while I do know how to manipulate power towers to a certain ...
31
votes
3answers
1k views

Is it true that this function $f(n)=n^{13}$?

Assume strictly monotone increasing function; such that $f:N^{+}\to N^{+}$, $h$ for all $n\in N^{+}$, $$f(f(f(n)))=f(f(n))\cdot f(n)\cdot n^{2015}$$ Prove or disprove:$f(n)=n^{13}$ ...
30
votes
10answers
6k 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 ...
30
votes
9answers
12k 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 ...
29
votes
2answers
1k views

How prove this function $f(x)=x!-x^n$ is injective

Question: For any positive integer $n$ such $n\neq 2^m-1, n\ge 2$, and function $f$ defined by $$f(x)=x!-x^n$$ show that : $f:N\to Z$ is injective. My idea: maybe for $x\neq y$ with $x,y\in N^{+}$, ...
28
votes
7answers
581 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?
28
votes
3answers
2k views

Why isn't the inverse of the function $x\mapsto x+\sin(x)$ expressible in terms of “the functions one finds on a calculator”?

The function $f(x)=x+\sin(x)$ is easily checked to be a bijection from the reals to itself, and so it has a unique inverse $y\mapsto g(y)$ such that $f\circ g=g\circ f$ are both the identity map. ...
27
votes
10answers
11k 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 ...
27
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 $f(n+1)=f(n)+m$, ...
26
votes
16answers
3k views

Produce unique number given two integers

Given two integers, $a$ and $b$, I need an operation to produce a third number $c$. This number does not have to be an integer. The restrictions are as follows: $c$ must be unique for the inputs ...
26
votes
5answers
2k views

Is there a function whose inverse is exactly the reciprocal of the function, that is $f^{-1} = \frac{1}{f}$?

Is there a function whose inverse is exactly the reciprocal of the function? That is $f^{-1} = \frac{1}{f}$. We know that the inverse of a function is not necessarily equal to its reciprocal in ...
25
votes
6answers
2k views

Is it possible to describe the Collatz function in one formula?

This is related to Collatz sequence, which is that $$C(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+1 & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$ Is it possible to describe ...
25
votes
0answers
382 views

What is that thing that keeps showing in papers on different fields?

A few months ago, when I was studying strategies for the evaluation of functional programs, I found that the optimal algorithm uses something called Interaction Combinators, a graph system based on a ...
24
votes
6answers
1k views

IMO 1987 - function

Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $f(f(n))=n+1987, \ \forall n \in \mathbb{N}$.
24
votes
10answers
10k views

Operator vs function

Could someone please explain the MATHEMATICAL difference between an operator and a function? I am not talking about these in terms of coding but rather the mathematical difference. Is operator also a ...
23
votes
3answers
2k views

Given $f(f(x))$ can we find $f(x)$?

Given $f(f(x))=x+2$ does it necessarily follow that $f(x)=x+1$? This question comes from a precalculus algebra student.
23
votes
6answers
716 views

Is there a function with this property?

Is there a real function over the real numbers with this property $\ \sqrt{|x-y|} \leq |f(x)-f(y)|$ ? My guess is no but can anyone tell me why? This came up as a question of one of my collegues and ...
22
votes
12answers
8k views

How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation. What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than ...
22
votes
7answers
2k 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 ...
22
votes
6answers
4k 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| $
22
votes
12answers
4k views

What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
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} ...
22
votes
4answers
2k views

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
22
votes
7answers
24k 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 ...
22
votes
4answers
8k 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?
22
votes
2answers
592 views

How to show that a root of the equation $x (x+1)(x+2) … (x+2009) = c $ can have multiplicity at most 2?

How to show that a root of the equation $$x (x+1)(x+2) ....... (x+2009) = c $$ can have multiplicity at most 2 , and to find the value of $ c $ for which this is possible. I proceeded by using the ...
22
votes
3answers
1k views

What's the name of this theorem?

It happens very often in physics that we find relations like: $$\int_V f(x) dx = \int_V g(x) dx$$ for an arbitrary volume $V$. From this we usually say "Since the volume is arbitrary, the integrands ...
22
votes
2answers
480 views

Calculate $\sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \ldots $

$$ \sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}+ \frac{1}{2}\sqrt{\frac{1}{2}}}} \times\ldots$$ I already know ...
22
votes
1answer
495 views

Peculiar locations of the root and the maximum of $(x+1)^{x+1}-x^{x+2}$

Related to some other problems, I got interested in this function: $$(x+1)^{x+1}-x^{x+2}$$ Its root is very close to $\pi$. This is Mathematica code that finds the root: ...
21
votes
5answers
5k views

Why can a circle be described by an equation but not by a function?

In high-school math functions always looked to me just like glorified equations. The only time I saw a meaningful difference was when we covered the equation of a circle and I realized that an ...
21
votes
9answers
7k views

Why is an ellipse, hyperbola, and circle not a function?

I am aware of the vertical line test. If you place a vertical line over a shape, and if it crosses more than once, it fails the vertical line test and is no longer a function. But I don't understand ...
21
votes
7answers
4k views

How to make a “function”?

I dropped out of school early when I was still a teenager and now I'm trying to take my GED. I'm really close to passing but I'm still having trouble understanding some concepts. In the pre-test, ...