All Questions
7
questions
2
votes
2
answers
59
views
Is there a 'simple' function that flips the order of positive numbers without making them negative?
If I want to flip the order of some numbers, I can just multiply them with -1. But is there a not too complicated way to do it such that the numbers remain positive?
Here's my attempt to word the ...
1
vote
1
answer
99
views
If $f$ is a function such that $f(f(x))=x^{2}-1$ determine the function $f(f(f(f(x))))$
I have tried $f(f(f(f(x))))=f\left(f\left(x^{2}-1\right)\right)$ . Since we know that $f(f(x))=x^{2}-1$, we have
$$
\begin{aligned}
f\left(f\left(x^{2}-1\right)\right) &=\left(x^{2}-1\right)^{2}-1 ...
- 55
7
votes
2
answers
312
views
What is a condition for two real functions $f,g$ to "commute", so $f(g(x))=g(f(x))$?
Say I'm given two functions $f,g$. Can I tell if they "commute" without actually trying them in the formula
$f(g(x))=g(f(x))$?
And given a function $f$, is there a way to find all functions $...
- 405
0
votes
1
answer
105
views
Suppose that for every $x$, $y$ such that $x$ is not equal to $y$ we have $|f(x) − f(y)| < |x − y|$
Let $a$ and $b$ two real numbers such that $a < b$ and $f : [a, b] \to [a, b]$.
Suppose that for every $x$, $y$ such that $x$ is not equal to $y$ we have
$|f(x) − f(y)| < |x − y|$. Show that ...
- 27
0
votes
1
answer
381
views
Linear Mean Function
I've been looking at functions $f:\mathbb{R}^n \to \mathbb{R}$ which necessarily satisfy the following 3 properties. Given $ a_1, a_2, \dots a_n \in \mathbb{R}^+ $
$\begin{array} { l l } 1. & f(...
user536517
1
vote
2
answers
314
views
Solving the functional equation $ xf(x) - yf(y) = (x - y)f(x + y) $ [duplicate]
I found the following functional equation:
Find all functions $f : \Bbb R \rightarrow \Bbb R $ such that:
$$ xf(x) - yf(y) = (x - y)f(x + y) \text{ for all }x, y \in \mathbb R $$
Could you please ...
- 870
8
votes
2
answers
362
views
Interesting functional equation: $f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$
Solve for the function f(x):
$$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$
I'm not able to solve this.
[For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me ...
- 5,243