How do you find the domain of #y=sqrt(x^2 - 6 x + 5)#?

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Answer 1 · 2017-07-30T05:03:13

The domain is #x in (-oo,1] uu [5,+oo)#

#y=sqrt(x^2-6x+5)#

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

Therefore,

#x^2-6x+5>=0#

We factorise the inequality

#(x-1)(x-5)>=0#

Let #f(x)=(x-1)(x-5)#

We build a sign chart

#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaaa)##1##color(white)(aaaaaaaaa)##5##color(white)(aaaaaaa)##+oo#

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

#color(white)(aaaa)##x-5##color(white)(aaaa)##-##color(white)(aaaa)##0##color(white)(aaa)##-##color(white)(aaa)##0##color(white)(aaaa)##+#

#color(white)(aaaa)##f(x)##color(white)(aaaaa)##+##color(white)(aaaa)##0##color(white)(aaa)##-##color(white)(aaa)##0##color(white)(aaaa)##+#

Therefore,

#f(x)>=0#, when #x in (-oo,1] uu [5,+oo)#

graph{sqrt(x^2-6x+5) [-10, 10, -5, 5]}