Find the inverse of $f(x) = (x-1)/(x+1)$ and then find $(f@f^-1)(x)$ ?

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Answer 1 · 2018-06-20T23:01:17

$f^-1(x)=(x+1)/(1-x)$

$(f@f^-1)(x)=x$

$f(x)=(x-1)/(x+1)=(x+1-2)/(x+1)=(x+1)/(x+1)+(-2)/(x+1)=1-2/(x+1)$

Let $y=f(x)$

$y=1-2/(x+1)$

$2/(x+1)=1-y$

$2=(1-y)(x+1)$

$2=x(1-y)+(1-y)$

$x(1-y)=2+y-1=y+1$

$x=(y+1)/(1-y)$

Thus $f^-1(x)=(x+1)/(1-x)$

By definition, we know that $(f@f^-1)(x)=x$

Find the inverse of f(x) = (x-1)/(x+1)f(x) = (x-1)/(x+1) and then find (f@f^-1)(x)(f@f^-1)(x)?