Question

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

Answer 1

Given function is `g(x)=\sqrt{-3x-2}`

Now, one thing for sure is that we can't have a real valued function involving imaginary outputs. So that means the values under the square root should be positive.

So, from the function itself `\sqrt{-3x-2}=0` is the least possible value. Squaring and rearranging on both sides we get
`-3x=2\impliesx=(-2)/3`

But what side of the real numbered line should be included? Left side of `-2/3` or the right?
So again, we take the square root, and then square and rearrange, but this time, we say that the function inside the square root should be positive.

In a more simple way of saying it mathematically
`-3x-2>0\implies-3x>2\impliesx<-2/3` (I've multiplied the inequality by a negative value, that's why the sign reverses).

So in the end, `x` belongs to the real number set, containing the values `(oo,(-2)/3]`