# Consider the function f(x)=1/(x-2), then how do you simplify f(f^-1(x))?

Aug 19, 2016

${f}^{- 1} \left(x\right) = \frac{1}{x} + 2$ and
$f \left({f}^{- 1} \left(x\right)\right) = x$

#### Explanation:

Given $f \left(x\right) = \frac{1}{x - 2}$ then

$f \left(g \left(x\right)\right) = \frac{1}{g \left(x\right) - 2}$. Now, if $f \left(g \left(x\right)\right) = x$ then

$g \left(x\right) = {f}^{- 1} \left(x\right)$

but

$\frac{1}{g \left(x\right) - 2} = x$ gives $g \left(x\right) = \frac{1}{x} + 2$

then

${f}^{- 1} \left(x\right) = \frac{1}{x} + 2$