Question

What is the domain of `sqrt((x^2-x-6)/(x-4)`?

Answer 1

The domain is `x in [-2,3] uu (4,+oo)`

Explanation:

The conditions are

`((x^2-x-6)/(x-4))>=0` and `x!=4`

Let `f(x)=((x^2-x-6)/(x-4))=((x+2)(x-3))/(x-4)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaaaaaa)` `3` `color(white)(aaaaaaa)` `4` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaaaa)` `-` `color(white)(aaaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaaaa)` `-` `color(white)(aaaaaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaaa)` `+` `color(white)(aa)` `0` `color(white)(aa)` `-` `color(white)(aa)` `||` `color(white)(aa)` `+`

Therefore,

`f(x)>=0` when `x in [-2,3] uu (4,+oo)`

graph{sqrt((x^2-x-6)/(x-4)) [-12.66, 19.38, -6.05, 9.99]}