#f(x)=x^2/(1-x^2)=x^2/((1-x)(1+x))#
As we cannot divide by #O#, #x!=1# and #x!=-1#
The domain of #f(x)# is #D_f(x)=RR-{-1,1}#
To calculate the range, we need to calculate #f^-1(x)#
Let #y=x^2/(1-x^2)#
We interchange #y# and #x#
#x=y^2/(1-y^2)#
Now, we calculate #y# in terms of #x#
#x(1-y^2)=y^2#
#x-xy^2=y^2#
#y^2(x+1)=x#
#y^2=x/(x+1)#
#y=sqrt(x/(x+1))#
The domain of #y# is the range of #f(x)#
What is underneath the #sqrt# sign is #>=0#
Therefore,
#x/(1+x)>=0#
We build a sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaaa)##-1##color(white)(aaaaaaaa)##0##color(white)(aaaa)##+oo#
#color(white)(aaaa)##x##color(white)(aaaaaaaaa)##-##color(white)(aaaa)##||##color(white)(aaaa)##-##color(white)(aaaa)##+#
#color(white)(aaaa)##x+1##color(white)(aaaaaa)##-##color(white)(aaaa)##||##color(white)(aaaa)##+##color(white)(aaaa)##+#
#color(white)(aaaa)##f^-1(x)##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aaaa)##-##color(white)(aaaa)##+#
Therefore,
#f^-1(x)>=0# when #x in (-oo,-1) uu [0,+oo)#
