Search found 550 matches
- Fri Mar 09, 2012 9:34 pm
- Forum: Algebra
- Topic: Functional equation(exhausted!!)
- Replies: 0
- Views: 1304
Functional equation(exhausted!!)
Find all function $f:R\rightarrow R$ such that $f(x+y)=f(x)+f(y)+2(xy+1)$.
Roots
If the equation $ax^{2}+(c-b)x+(e-d)=0$ has real roots greater than $1$,then show that
$ax^{4}+bx^{3}+cx^{2}+dx+e=0$
has at least one real root.
$ax^{4}+bx^{3}+cx^{2}+dx+e=0$
has at least one real root.
- Wed Mar 07, 2012 12:26 am
- Forum: Algebra
- Topic: Functional equation
- Replies: 3
- Views: 2592
Functional equation
Find all functions $f:Z\rightarrow R$ such that
$f(\frac{x+y}{3})=\frac{f(x)+f(y)}{2}$
Surely $x+y$ is divisible by $3$
$f(\frac{x+y}{3})=\frac{f(x)+f(y)}{2}$
Surely $x+y$ is divisible by $3$
- Sun Mar 04, 2012 4:31 pm
- Forum: Algebra
- Topic: Functional equation
- Replies: 1
- Views: 1766
Functional equation
Find all strictly increasing functyions $f$ determined from real to real such that
$f(x)+g(x)=2x$,
where $g$ is the inverse of $f$.
NOTE:Can you tell me how to express ''from real to real'' inside $$?
$f(x)+g(x)=2x$,
where $g$ is the inverse of $f$.
NOTE:Can you tell me how to express ''from real to real'' inside $$?
- Sun Mar 04, 2012 4:15 pm
- Forum: Algebra
- Topic: AN INTERESTING PROBLEM BY SAKAL DA
- Replies: 5
- Views: 4379
Re: AN INTERESTING PROBLEM BY SAKAL DA
upssssss........sorrry..........got the mistake probably
- Sun Mar 04, 2012 4:07 pm
- Forum: Algebra
- Topic: funtional equation [really hard (maybe :? )]
- Replies: 9
- Views: 4837
Re: funtional equation [really hard (maybe :? )]
Setting $x=0$,$y=0$,we get $2f(0)^{2}+f(0)=0$.Hence,$f(0)=0$ or $-\frac{1}{2}$.Setting $y=0$ and using the later case,we deduce that , $-f(x)-\frac{1}{2}=0$ for all real $x$ that is,$f(x)=-\frac{1}{2}$ is a solution. Now first set $y=x$,then $y=-x$.It implies that $f(2x^{2})=2f(x^{2})+2f(x)^{2}=2f(x...
- Sun Mar 04, 2012 3:40 pm
- Forum: Algebra
- Topic: Functional equation
- Replies: 7
- Views: 4107
Re: Functional equation
Yeah,I am so sorry. I wanted to write $a^{2}+a=1$
- Sat Mar 03, 2012 11:49 pm
- Forum: Algebra
- Topic: Functional equation
- Replies: 10
- Views: 6352
Re: Functional equation
I doubt if such a non-constant function really exists.Plugging in $-1,1,0$ for $f(x)$ says me that I am right,but I think you do have solutions in hand....will assure me so that I can retry ?
- Sat Mar 03, 2012 11:25 pm
- Forum: Algebra
- Topic: Functional equation
- Replies: 7
- Views: 4107
Re: Functional equation
$f(n)=\left\lfloor na\right\rfloor+1$
Where $a$ the greater root of $a^{2}=a+1$.Simply try to consider cases for $f(k)$ for some natural $k$.I won't tell the value of $k$
Where $a$ the greater root of $a^{2}=a+1$.Simply try to consider cases for $f(k)$ for some natural $k$.I won't tell the value of $k$
- Thu Mar 01, 2012 11:24 pm
- Forum: Algebra
- Topic: AN INTERESTING PROBLEM BY SAKAL DA
- Replies: 5
- Views: 4379
Re: AN INTERESTING PROBLEM BY SAKAL DA
The answer is:
$a_{2012}=2012+2^{2^{2011}}$
$a_{2012}=2012+2^{2^{2011}}$