How do you find the domain and range of #sqrt [x/(x-6)]#?

1 Answer
Apr 12, 2017

Answer:

The domain is #x in (-oo,0]uu(6,+oo)#
The range is #[0,1)uu(1,+oo)#

Explanation:

To find the domain, what's under the square root sign is #>=0#

So,

#x/(x-6)>=0#

Let #p(x)=x/(x-6)#

We need to build a sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##0##color(white)(aaaaaaaa)##6##color(white)(aaaaaaaa)##+oo#

#color(white)(aaaa)##x##color(white)(aaaaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aaaa)##+#

#color(white)(aaaa)##x-6##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aaaaa)##||##color(white)(aaaa)##+#

#color(white)(aaaa)##p(x)##color(white)(aaaaa)##+##color(white)(aaaa)##-##color(white)(aaaaa)##||##color(white)(aaaa)##+#

Therefore,

#p(x)>=0# when #x in (-oo,0]uu(6,+oo)#

The domain is #x in (-oo,0]uu(6,+oo)#

#lim_(x->+-oo)p(x)=lim_(x->+-oo)x/x=1#

When #x=0#, #p(x)=0#

#lim_(x->6^+)p(x)=+oo#

So,

the range is #[0,1)uu(1,+oo)#

graph{sqrt(x/(x-6)) [-25.67, 25.65, -12.83, 12.84]}