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

Nov 22, 2015

Domain: $\left(- \infty , + \infty\right) \in \mathbb{R}$
Range: $\left(- \infty , - 5\right] \in \mathbb{R}$

#### Explanation:

$F \left(x\right) = - 2 {\left(x + 3\right)}^{2} - 5$ can be evaluated for all values of $x \in \mathbb{R}$
so the Domain of $F \left(x\right)$ is all $\mathbb{R}$

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

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

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