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

1 Answer
Jan 26, 2017

Answer:

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)##x##color(white)(aaaa)##-oo##color(white)(aaaaaaa)##-4##color(white)(aaaaa)##1##color(white)(aaaaa)##+oo#

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

#color(white)(aaaa)##x-1##color(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[#