The function is
y=(2x^2)/(x^2-1)
We factorise the denominator
y=(2x^2)/((x+1)(x-1))
Therefore,
x!=1 and x!=-1
The domain of y is x in (-oo,-1) uu (-1,1) uu (1,+oo)
Let's rearrage the function
y(x^2-1)=2x^2
yx^2-y=2x^2
yx^2-2x^2=y
x^2=y/(y-2)
x=sqrt(y/(y-2))
For x to a solution, y/(y-2)>=0
Let f(y)=y/(y-2)
We need a sign chart
color(white)(aaaa)ycolor(white)(aaaa)-oocolor(white)(aaaaaa)0color(white)(aaaaaaa)2color(white)(aaaa)+oo
color(white)(aaaa)ycolor(white)(aaaaaaaa)-color(white)(aaa)0color(white)(aaa)+color(white)(aaaa)+
color(white)(aaaa)y-2color(white)(aaaaa)-color(white)(aaa)color(white)(aaa)-color(white)(aa)||color(white)(aa)+
color(white)(aaaa)f(y)color(white)(aaaaaa)+color(white)(aaa)0color(white)(aa)-color(white)(aa)||color(white)(aa)+
Therefore,
f(y)>=0 when y in (-oo,0] uu (2,+oo)
graph{2(x^2)/(x^2-1) [-16.02, 16.02, -8.01, 8.01]}
