Question

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

Answer 1

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

Explanation:

`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]}