How do you find the domain of #f(x) = sqrt ( x- (3x^2))#?

1 Answer
Mar 18, 2018

Answer:

The domain of #f(x)# is #x in [0,1/3]#

Explanation:

What's under the #sqrt()# sign is #>=0#

Here,

#f(x)=sqrt(x-3x^2)#

Therefore,

#x-3x^2>=0#

#x(1-3x)>=0#

Let #g(x)=x(1-3x)#

Build the sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaaa)##0##color(white)(aaaaaaa)##1/3##color(white)(aaaaaa)##+oo#

#color(white)(aaaa)##x##color(white)(aaaaaaaa)##-##color(white)(aaa)##0##color(white)(aaa)##+##color(white)(aaaaaa)##+#

#color(white)(aaaa)##1-3x##color(white)(aaaa)##+##color(white)(aaa)####color(white)(aaaa)##+##color(white)(aa)##0##color(white)(aaaa)##-#

#color(white)(aaaa)##g(x)##color(white)(aaaaaa)##-##color(white)(aaa)##0##color(white)(aaa)##+##color(white)(aa)##0##color(white)(aaaa)##-#

Therefore,

#g(x)<=0#, #=>#, #x in [0,1/3]#

graph{sqrt(x-3x^2) [-0.589, 1.096, -0.307, 0.536]}