Question

Find the domain (x2-1)/x2+1

Answer 1

Answer:

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)xcolor(white)(aaaa)-oocolor(white)(aaaaaa)-1color(white)(aaaaaaaa)0color(white)(aaaa)+oo

color(white)(aaaa)xcolor(white)(aaaaaaaaa)-color(white)(aaaa)||color(white)(aaaa)-color(white)(aaaa)+

color(white)(aaaa)x+1color(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)