Find the value of z

Posted January 18th, 2008 by alpha

Given

$x=\frac{y+1}{y-1},\: y =\frac{2z+1}{2z-1}$

express z in terms of x.Hence find the value of z when x=5

Comments

On January 18th, 2008 Structure says:

Let $f(x)=\frac{x+1}{x-1}$
then

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

Now
Given

$x=\frac{y+1}{y-1},\: y =\frac{2z+1}{2z-1}$

express z in terms of x.Hence find the value of z when x=5
we have $x=f(y),\:y=f(2z)$
$f(x)=f(f(y))=y,\:f(y)=f(f(2z))=2z$
So

$2z=f(y)=x$

and finally

$z=\frac{x}{2}$