Question

What is the domain of `h(x)=sqrt(( x- (3x^2)))`?

Answer 1

Domain: `(0, 1/3)`

Explanation:

Right from the start, you know that the domain of the function must only include values of `x` that will make the expression under the square root positive.

In other words, you need to exclude from the function's domain any value of `x` will result in

`x - 3x^2 < 0`

The expression under the square root can be factored to give

`x - 3x^2 = x * (1 - 3x)`

Make this expression equal to zero to find the values of `x` that make it negative.

`x * (1 - 3x) = 0 implies {(x = 0), (x = 1/3) :}`

So, in order for this expression to be positive, you need to have
`x>0` and `(1-3x) > 0`, or `x<0` and `(1-3x)<0`.

Now, for `x<0`, you have

`{(x<0), (1 - 3x > 0) :} implies x * (1-3x) < 0`

Likewise, for `x > 1/3`, you have

`{(x > 0), (1 - 3x > 0) :} implies x * (1-3x) < 0`

This means that the only values of `x` that will make that expression positive can be found in the interval `x in (0, 1/3)`.

Any other value of `x` will cause the expression under the square root to be negative. The domain of the function will thus be `x in (0, 1/3)`.

graph{sqrt(x-3x^2) [-0.466, 0.866, -0.289, 0.377]}