Elementary questions about functions, notation, properties, and operations such as function composition.
62
votes
6answers
2k views
Find a real function, $f$, s.t. $f(f(x)) = -x$.
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 function $f: ...
37
votes
2answers
538 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:
...
33
votes
7answers
2k 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 ...
33
votes
4answers
1k 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 ...
33
votes
4answers
2k 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 ...
30
votes
11answers
2k 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 ...
29
votes
4answers
799 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 ...
25
votes
3answers
373 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.
...
22
votes
6answers
593 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 ...
20
votes
2answers
381 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 ...
19
votes
9answers
2k 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 ...
18
votes
7answers
771 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 ...
17
votes
3answers
967 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 ...
17
votes
2answers
590 views
Largest circle between $y=x^n$ and $y=\sqrt[n]{x}$
Something I have been wondering about for a while. Let us look at the area between $x^n$ and $\sqrt[n]{x}$ when $x\in [0,1]$. Where $n$ is a positive integer.
Below is an image.
With a given n, how ...
16
votes
9answers
1k 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 ...
16
votes
4answers
641 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.
16
votes
2answers
627 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 $f(n+1)=f(n)+m$, ...
16
votes
1answer
541 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 ...
15
votes
3answers
529 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 ...
15
votes
5answers
467 views
IMO 1987 - function
Show that there is no function $f: \mathbb{N} \to \mathbb{N}$ such that $f(f(n))=n+1987, \ \forall n \in \mathbb{N}$.
15
votes
1answer
188 views
A bijection from the plane to itself that takes a circle to a circle must take a straight line to a straight line.
Let $ f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} $ be a bijective function. If the image of any circle under $ f $ is a circle, prove that the image of any straight line under $ f $ is a straight ...
15
votes
1answer
344 views
“Converse” of Taylor's theorem
Let $f:(a,b)\to\mathbb R$. We know that for every $c\in(a,b)$ we can write $f(t)=\sum_{i=0}^k a_i(c)(t-c)^i+o\left((t-c)^k\right)$ and $\forall i$ $a_i(c)$ is continuous (with respect to $c$). Can we ...
14
votes
8answers
3k 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 ...
14
votes
5answers
536 views
How to solve this equation:$\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-5}}}}=5$
Please, help me to solve this equation:
$$\sqrt{x-\sqrt{x-\sqrt{x-\sqrt{x-5}}}}=5$$
I think if we continue to square, it would have to face a high-order equation.
I guess $x = 30$, but do not know ...
14
votes
1answer
321 views
On Vanishing Riemann Sums and Odd Functions
Let $ f: [-1,1] \to \mathbb{R} $ be a continuous function. Suppose that the $ n $-th midpoint Riemann sum of $ f $ vanishes for all $ n \in \mathbb{N} $. In other words,
$$
\forall n \in ...
14
votes
1answer
349 views
prove that there is no function that
Prove that there is no function on open interval $(-1,1)$, which has only
finite number of discontinuity point, such that its graph is invariant
under rotation by the right angle around the origin.
13
votes
11answers
980 views
If $f(x)\leq f(f(x))$ for all $x$, is $x\leq f(x)$?
If I have $f(x)\leq f(f(x))$ for all real $x$, can I deduce $x\leq f(x)$?
Thank you.
13
votes
3answers
2k views
Is there a bijection between $(0,1)$ and $\mathbb{R}$ that preserves rationality?
While reading about cardinality, I've seen a few examples of bijections from the open unit interval $(0,1)$ to $\mathbb{R}$, one example being the function defined by $f(x)=\tan\pi(2x-1)/2$. Another ...
13
votes
3answers
2k views
How do you handle the floor and ceiling function in an equation?
I tried to do some math in a blog post of mine and came to one with a floor function. I wasn't sure how to deal with it so I just ignored it, and then added the ceiling function in my final equation ...
13
votes
4answers
437 views
Example of a rational function such that : $(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$
Can any one give me example of: rational functions $f, g$ and $h$ with rational coefficients such that
$$(f(x))^{3} + (g(x))^{3} + (h(x))^{3}=x$$
Also, if anyone knows a procedure for constructing ...
13
votes
2answers
336 views
Increasing orthogonal functions
What is the maximal $n$ such that there exist functions $f_1, \dots, f_n:[0,1] \to \mathbb{R}$ that are all bounded, non-decreasing, and mutually orthogonal in $L^2([0,1])$?
13
votes
0answers
199 views
What is the algebraic structure of functions with fixed points?
So I just noticed that the set of functions with a fixed point
$$f(x_0)=x_0,$$
are closed under composition
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
12
votes
11answers
3k views
Piecewise functions: Got an example of a real world piecewise function?
Looking for something beyond a contrived textbook problem concerning jelly beans or equations that do not represent anything concrete. Not just a piecewise function for its own sake. Anyone?
12
votes
6answers
2k views
Can there be a function that's even and odd at the same time?
I woke up this morning and had this question in mind. Just curious if such function can exist.
12
votes
7answers
600 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 ...
12
votes
3answers
1k views
Domain, Co-Domain & Range of a Function
I'm a little confused between the difference between the range & co-domain of a function. Are they not the same thing (i.e. all possible outputs of the function)?
12
votes
3answers
420 views
Name for $(1-x)$?
The multiplicative inverse of $x$ is $\frac{1}{x}$,
and the additive inverse of $x$ is $-x$,
is there a similar term for $(1-x)$?
12
votes
2answers
296 views
A curve that starts with exponential growth and levels out
I'm stuck on a mathematical problem in a small feature I'm building for a website. I need a function that starts out with exponential growth and than levels out as x grows. I've drawn this wonderful ...
12
votes
2answers
521 views
isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?
Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$.
Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
12
votes
2answers
625 views
Inverse function of $y=\frac{\ln(x+1)}{\ln x}$
I've been wondering for a while if it's possible to find the inverse function of $y=\frac{\ln(x+1)}{\ln x}$ over the reals. This is the same as finding the positive real root of $x^y-x-1$. I realize ...
11
votes
6answers
921 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
5answers
915 views
can any continuous function be represented as a sum of convex and concave function?
I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave.
There could be ...
11
votes
2answers
351 views
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
11
votes
4answers
313 views
$f(16x)=16f(x) $ and $ f$ is continuous
$f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function such that $f(16x)=16f(x)$ for every real $x$.
Should it be $f(x)=ax$? How can I prove that?
11
votes
2answers
116 views
Functions satisfying $f(m+f(n)) = f(m) + n$
I am a real newbie when it comes to funtions, and I don't understand what is supposed to happen or what I'm supposed to find when I get given an olympiad type question concerning functions. Could you ...
11
votes
2answers
175 views
Can every real function be represented as two shifted even functions?
I saw the theorem that every function can be represented as the sum of and even and odd function, and this made me wonder: can every function from the reals to the reals, defined on all the reals, be ...
11
votes
2answers
138 views
About the solution of the infinite recurrence $f(x,f(x,f(x,f(x,f(…))))=a$
On internet I found some recreational problems as $$3=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}$$ $$\frac{1}{2}=\frac{1}{x+\frac{1}{x+...}}$$ $$2=x^{x^{x^{...}}}$$ And the trick to solve them was just ...
10
votes
3answers
441 views
Is there any way to integrate $\frac1{f(x)}$ in terms of the integral of $f(x)$?
Is there any way to find $$\int \frac1{f(x)}\mathrm dx$$ in terms of $\int f(x) \mathrm dx$, $f(x)$ and its derivatives?
10
votes
5answers
361 views
What's the difference between arccos(x) and sec(x)
My question might sound dumb, but I don't really see why the graphics of arccos(x) and sec(x) are different, because as far as I know arccos is the inverse cosine function (cos(x)^-1) and sec equals ...
10
votes
3answers
457 views
If $f(x)f(y)=f(\sqrt{x^2+y^2})$ how to find $f(x)$
As we know, for the $$f(x)f(y)=f(x+y)$$ $f(x)=\mathrm e^{\alpha x}$ is a solution.
What about
$f(x)f(y)=f(\sqrt{x^2+y^2})$?
Does anybody know about the solution of the function equation?
I tried ...
