Period : 2

**dhiab** · November 5th 2009, 08:43 PM

Find the period for the function f in R:
f(x+1)f(x)+f(x+1)+1=f(x)

**red_dog** · November 6th 2009, 01:27 AM

We have $f(x+1)=\frac{f(x)-1}{f(x)+1}$ .

Replace x with x+1:

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

Replace x with x+2:

$f(x+4)=-\frac{1}{f(x+2)}=-\frac{1}{\displaystyle -\frac{1}{f(x)}}=f(x)$ .

Then the period is 4.

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