Period

dhiab · #1 $Old$ June 18th, 2009, 01:40 PM

function , is a real number .

1 ) Find the period of this function
2 ) Give the exempl for .

pickslides · #2 $Old$ June 18th, 2009, 03:42 PM

Image does not show, happens with a lot of your posts mate!

VonNemo19 · #3 $Old$ June 18th, 2009, 03:46 PM

Quote:

Originally Posted by dhiab $View Post$

function , is a real number .

1 ) Find the period of this function
2 ) Give the exempl for .

This doesn't make any sense.

pankaj · #4 $Old$ June 18th, 2009, 08:02 PM

The correct question is as follows:
If $f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^2}$
then prove that
$f(x+2a)=f(x)$
(Moscow Olympiad,1950)

Solution:

Replace $x$ by $x+a$ in the above expression

$f(x+a+a)=\frac{1}{2}+\sqrt{f(x+a)-(f(x+a))^2}$

$f(x+2a)=\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt{f(x)-(f(x))^2}-\left(\frac{1}{2}+\sqrt{f(x)-(f(x))^2}\right)^2}$

$f(x+2a)=\frac{1}{2}+\sqrt{\frac{1}{4}-f(x)+(f(x))^2}=\frac{1}{2}+\sqrt{\left(f(x)-\frac{1}{2}\right)^2}$

$f(x+2a)=\frac{1}{2}+f(x)-\frac{1}{2}$

$f(x+2a)=f(x)$

There are two more questions of this type:

(i)If $f(x-1)+f(x+1)=f(x)$ ,then find one of the periods of $f(x)$ .

(ii)If $f(x-1)+f(x+1)=\sqrt{3}f(x)$ ,then find one of the periods of $f(x)$

dhiab · #5 $Old$ June 19th, 2009, 12:22 AM

Quote:

Originally Posted by VonNemo19 $View Post$

This doesn't make any sense.

Hello : exo :
(Moscow Olympiad,1950)
watch follow the sulution ci over Thank you

Quote:

Originally Posted by pankaj $View Post$

The correct question is as follows:
If $f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^2}$
then prove that
$f(x+2a)=f(x)$
(Moscow Olympiad,1950)

Solution:

Replace $x$ by $x+a$ in the above expression

$f(x+a+a)=\frac{1}{2}+\sqrt{f(x+a)-(f(x+a))^2}$

$f(x+2a)=\frac{1}{2}+\sqrt{\frac{1}{2}+\sqrt{f(x)-(f(x))^2}-\left(\frac{1}{2}+\sqrt{f(x)-(f(x))^2}\right)^2}$

$f(x+2a)=\frac{1}{2}+\sqrt{\frac{1}{4}-f(x)+(f(x))^2}=\frac{1}{2}+\sqrt{\left(f(x)-\frac{1}{2}\right)^2}$

$f(x+2a)=\frac{1}{2}+f(x)-\frac{1}{2}$

$f(x+2a)=f(x)$

There are two more questions of this type:

(i)If $f(x-1)+f(x+1)=f(x)$ ,then find one of the periods of $f(x)$ .

(ii)If $f(x-1)+f(x+1)=\sqrt{3}f(x)$ ,then find one of the periods of $f(x)$

Hello : Thank you for correction