How do you find the domain and range of f(x) = sqrt{(x - 1)/(x + 4)}?

1 Answer
Jan 26, 2017

The domain is D_f(x)=x in ]-oo,-4[ uu [1,+oo[
The range is R_f(x)=[0, +oo[

Explanation:

As you cannot divide by 0, x!=-4

and what's under the sqrt sign is >=0

(x-1)/(x+4)>=0

Let g(x)=(x-1)/(x+4)

We build a sign chart

color(white)(aaaa)xcolor(white)(aaaa)-oocolor(white)(aaaaaaa)-4color(white)(aaaaa)1color(white)(aaaaa)+oo

color(white)(aaaa)x+4color(white)(aaaaaa)-color(white)(aaaa)||color(white)(aa)+color(white)(aaaa)+

color(white)(aaaa)x-1color(white)(aaaaaa)-color(white)(aaaa)||color(white)(aa)-color(white)(aaaa)+

color(white)(aaaa)g(x)color(white)(aaaaaaa)+color(white)(aaaa)||color(white)(aa)-color(white)(aaaa)+

Therefore,

g(x)>=0 when x in ]-oo,-4[ uu [1,+oo[

The domain is D_f(x)=x in ]-oo,-4[ uu [1,+oo[

lim_(x->+-oo)g(x)=lim_(x->+-oo)x/x=lim_(x->+-oo)1=1

lim_(x->-4^-)g(x)=lim_(x->-4^-)-5/(0^-) = +oo

g(1)=0/5=0

So,

the range is R_f(x)=[0, +oo[