You must have copied it incorrectly.
As written it is not one-to-one.
$\displaystyle g\left( 3 \right) = g\left( {\frac{{ - 3}}
{5}} \right) = \frac{{ - 3}}
{2}$
In that case they cannot have different signs.
Note $\displaystyle \left| x \right| < 1\; \Rightarrow \;1 - \left| x \right| > 0$.
So the numerators must have the same sign.