Question

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

Answer 1

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]}