As you cannot divide by 0, the denominator is !=0
4-x^2!=0
(2+x)(2-x)!=0
=>, x!=-2 and x!=2
Therefore,
The domain is x in (-oo,-2)uu(-2,2)uu(2,+oo)
Let
y=(3+x^2)/(4-x^2)
y(4-x^2)=3+x^2
4y-yx^2=3+x^2
x^2(1+y)=4y-3
x^2=(4y-3)/(1+y)
x=sqrt((4y-3)/(1+y))
Therefore,
(4y-3)/(1+y)>=0
=>, y!=-1
Let g(y)=(4y-3)/(1+y)
We build a sign chart
color(white)(aaaa)ycolor(white)(aaaa)-oocolor(white)(aaaaaa)-1color(white)(aaaaaaa)3/4color(white)(aaaaaaa)+oo
color(white)(aaaa)1+ycolor(white)(aaaaaa)-color(white)(aaa)||color(white)(aaaa)+color(white)(aaaaaaa)+
color(white)(aaaa)4y-3color(white)(aaaaa)-color(white)(aaa)||color(white)(aaaa)-color(white)(aa)0color(white)(aaaa)+
color(white)(aaaa)g(y)color(white)(aaaaaaa)+color(white)(aaa)||color(white)(aaaa)-color(white)(aa)0color(white)(aaaa)+
Therefore,
g(y)>=0 when y in (-oo,-1)uu [3/4,+oo)
graph{(3+x^2)/(4-x^2) [-12.66, 12.65, -6.33, 6.33]}
