Question

What is the domain and range of `F(x) = -2(x + 3)² - 5`?

Answer 1

Domain: `(-oo,+oo) in RR`
Range: `(-oo,-5] in RR`

Explanation:

`F(x) = -2(x+3)^2-5` can be evaluated for all values of `x in RR`
so the Domain of `F(x)` is all `RR`

`-2(x+3)^2-5`
is a quadratic in vertex form with vertex at `(-3,-5)`
and the negative coefficient of `(x+3)^2` tells us that the quadratic opens downward;
therefore `(-5)` is a maximum value for `F(x)`

Alternative way of seeing this:
`(x+3)^2` has a minimum value of `0` (this is true for any squared Real value)
therefore
`-2(x+3)^2` has a maximum value of `0`
and
`-2(x+3)^2-5` has a maximum value of `(-5)`

Second alternative
consider the graph of this function:
graph{-2*(x+3)^2-5 [-17.42, 5.08, -9.78, 1.47]}