Show that \(f(f(x))=x\)

Solution

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

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

Simplify the numerator: $$ \frac{x+1}{x-1}+1 = \frac{x+1+x-1}{x-1} = \frac{2x}{x-1} $$

Simplify the denominator: $$ \frac{x+1}{x-1}-1 = \frac{x+1-x+1}{x-1} = \frac{2}{x-1} $$

Therefore, $$ f(f(x)) = \frac{\frac{2x}{x-1}}{\frac{2}{x-1}} $$

$$ =\frac{2x}{x-1}\times\frac{x-1}{2} $$

$$ =x $$

Hence, $$ \boxed{f(f(x))=x} $$