function

$f(f(f(5)))=$$\frac{\left(\frac{\left(\frac{\left(x-1\right)}{\left(x+1\right)}-1\right)}{\left(\frac{\left(x-1\right)}{\left(x+1\right)}+1\right)}-1\right)}{\left(\frac{\left(\frac{\left(x-1\right)}{\left(x+1\right)}-1\right)}{\left(\frac{\left(x-1\right)}{\left(x+1\right)}+1\right)}+1\right)}$

That simplifies to $\frac{-x-1}{x-1}$

Substituting 5 gives $\frac{-\left(5\right)-1}{\left(5\right)-1}$

=$-3/2$

Just do it one bit at a time.

Let f(x) = (x - 1)/(x + 1). Compute f(f(f(5))).

f(5)=(5 - 1)/(5 + 1) = 4/6  =  2/3

$f(f(5)) = f(\frac{2}{3}) \\ =\frac{ (\frac{2}{3}-1) }{ (\frac{2}{3}+1) }\\~\\ =\frac{ \frac{-1}{3} }{ \frac{5}{3} }\\~\\ =\frac{-1}{5}\\~\\ f(f(f(5)) )=f( f(\frac{2}{3}))=f(\frac{-1}{5}) \\ etc$

you need to check what i have done and then finish it.