Question

How do you find the domain of `sqrt{-x - 2}`?

Answer 1

All values of x smaller than or equal to -2 (or `x<=-2`)

Explanation:

Domain of `sqrt(f(x))` is always all values of x for which `f(x)>=0` as if `f(x)` is negative, `sqrt(f(x))` is no longer real.

Thus, here `f(x)=-x-2`

`f(x)>=0` which means that `-x-2>=0`

Adding 2 on both sides, we get
`-x>=2`

If we multiply -1 on both sides, we'll have to reverse the sign of inequality, so this becomes

`x<=-2`

Hence for all values of x smaller than or equal to -2, `sqrt(-x-2)` will be real and hence, these values are the domain for this function.