All Questions
11
questions
7
votes
2
answers
159
views
Find all real functions that satisfy: $f(x+y) = (f(x))^2 + (f(y))^2$, $x$, $y$ $\in$ $\mathbb{R}$
I am trying to find all real functions that satisfy the property: $f(x+y) = (f(x))^2 + (f(y))^2$, $x$, $y$ real numbers. I tried to substitute $x$ and $y$ with $0$ but end up with nothing, then I ...
- 301
7
votes
2
answers
551
views
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ where $f(xf(y)+f(x)+y)=xy+f(x)+f(y)$
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ for two real numbers $x$ and $y$ where $f(xf(y)+f(x)+y)=xy+f(x)+f(y)$
For $x=0$ and $y=-f(0)$ then $f(-f(0))=0$. So, there is a real root $r_0$ ...
- 351
0
votes
0
answers
47
views
Find $f(3)$ if $f(f(x) - y) = f(x) + f(f(y) - f(-x)) + x$ [duplicate]
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
$$f(f(x) - y) = f(x) + f(f(y) - f(-x)) + x$$for all real numbers $x$ and $y.$
Let $n$ be the number of possible values of $f(3),$ and let $s$...
- 800
6
votes
2
answers
367
views
How to strengthen $ h \big( 2 h ( x ) \big) = h ( x ) + x $ to force $ h $ to be linear?
Let $ h : \mathbb R \to \mathbb R $ be an injective function such that
$$
h \big( 2 h ( x ) \big) = h ( x ) + x
$$
for all $ x \in \mathbb R $, and $ h ( 0 ) = 0 $. What would be an as mild as ...
- 12.1k
0
votes
1
answer
192
views
How to find all the continuous functions satisfying an equation? [duplicate]
The problem that I want to solve is:
"find all the continuous functions $f\colon \mathbb R\to \mathbb R$ such that for every $x$, $f(f(f(x))) = x $ , I know that f(x) = x is an answer but how can ...
- 540
2
votes
1
answer
88
views
Find all function $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ such that : $f(ax)f(by)=f(ax+by)+cxy$ where $a,b,c>0$
If $f:\ \Bbb{R}\ \longrightarrow\ \Bbb{R}$ and $a,b,c>0$, then find all function such that :
$$f(ax)f(by)=f(ax+by)+cxy,\quad \text{where } a,b,c>0 \text{ for all } x,y\in \Bbb{R}.$$
My attempt
...
user664780
2
votes
2
answers
110
views
Is this theorem true?
If $f(x)+f(y)=f(x+y)$, then:
$f(x)=a x$
where $a$ is a constant.
Is the above statement true? Is there a way of proving it?
The application of this theorem is in the last part of page 52 (second ...
- 1,141
14
votes
2
answers
436
views
Solve $f(x+f(2y))=f(x)+f(y)+y$
Find all $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for each $x$ and $y$ in $\mathbb{R}^+$, $$f(x+f(2y))=f(x)+f(y)+y$$
Note:
$f(x)=x+b$ is a solution for all $b\in\mathbb{R}^+$ but I can not prove ...
1
vote
2
answers
316
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 ...
- 5,263
0
votes
1
answer
1k
views
Prove an additive function has property f(x)=x
So I am given a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(x+y)=f(x)+f(y)$ for all real $x$ and $y$, is continuous at $x=0$, and $f(1)=1$. I need to show that $f(x)=x$ for all real $...
- 445