Question

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

Answer 1

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]}