Linked Questions

89
votes
5answers
22k views

Is infinity an odd or even number?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.
45
votes
8answers
5k 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 ...
4
votes
3answers
813 views

Is the cardinality of the set of rational numbers, $\mathbb{Q}$ odd?

Let $\mathbb{Q}$ be the set of unique rational numbers of the form $m/n$ where $m\in\mathbb{Z}$ and $n\in\mathbb{N}$. Define the following two sets: $$\begin{align} \mathcal{A}&=\{m/n\ \vert\ ...
6
votes
1answer
688 views

Solutions of $f(f(z)) = e^z$

It is my impression that if we find a function f(z) that satisfies $$f(f(z)) = e^z $$ there is only one point z that satisfies the relation. This dawned on me when I noticed that the pesky z that ...
13
votes
0answers
524 views

Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
4
votes
1answer
197 views

There exist a function such that $f\circ f(x)=e^x$? [duplicate]

Based on this question: How to calculate $f(x)$ in $f(f(x)) = e^x$? I would like to know if I can get a function such that $f:\mathbb R \to \mathbb R^+$, defined by $f\circ f(x)=e^x$. My guess is no, ...
6
votes
2answers
236 views

Roots of maps of finite sets

Let $X_n$ be a set with $n$ elements. Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself. We give it the operation of composition. I am curious if there is a nice formula for the following ...
2
votes
0answers
426 views

Solving $f(f(x))=g(x)$ equations [duplicate]

Possible Duplicate: Square root of a function (in the sense of composition) I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known function. ...
7
votes
0answers
130 views

Any chance for a solution of $f[f(x)]=x^2+1$? [duplicate]

Not sure if this is a closed or open question. But the question is suppose $f[f(x)]=x^2+1$ then what is $f(x)$? Though the question does not refer to the domain let us suppose it is defined on $\Bbb R$...
3
votes
1answer
66 views

Apply function fractional times

For example, one can apply $\cos x$ to number $a$ one time to get $\cos a$, two times $\cos \cos a$, three times $\cos \cos \cos a$, and so on. Is there a way to define fractional application for $\...