How do you find the domain of the function f#(x)=sqrt(x^3-64 x) #?

1 Answer
May 19, 2017

Answer:

The domain of #f(x)# is #x in [-8,0] uu [8,+oo)#

Explanation:

What's under the square root sign is #>=0#

So,

#x^3-64x>=0#

#x^3-64x=x(x^2-64)=x(x+8)(x-8)#

Let #g(x)=x(x+8)(x-8)>=0#

We build a sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaa)##-8##color(white)(aaaa)##0##color(white)(aaaaa)##+8##color(white)(aaaaa)##+oo#

#color(white)(aaaa)##x+8##color(white)(aaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(aaaaa)##+#

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

#color(white)(aaaa)##x-8##color(white)(aaaaa)##-##color(white)(aaaa)##-##color(white)(aaaa)##-##color(white)(aaaaa)##+#

#color(white)(aaaa)##g(x)##color(white)(aaaaaa)##-##color(white)(aaaa)##+##color(white)(aaaa)##-##color(white)(aaaaa)##+#

Therefore,

#g(x)>=0# when #x in [-8,0] uu [8,+oo)# graph{sqrt(x^3-64x) [-27.54, 30.2, -4.14, 24.74]}