Learning with an old russian math book, i found the following evaluation for the function $f(x)=\sqrt{1+x^2}$:

$f(\frac1x)=\vert x \vert^{-1}\sqrt{1+x^2}$

My evaluation gave me 

$\sqrt{1+\dfrac1{x^2}}$

Are these two statements algebraically equivalent? I don't see the connection to the absolute values here.

Any help is appreciated.