# What is the range of f(x)=–5-2(x+3)^2?

Dec 31, 2016

The range is $\left\{y | y \le \text{-5}\right\}$.

#### Explanation:

The range of a function is simply all the possible outputs that function can give.

Mathematically speaking, a number $y$ is in the range of a function $f$ when we can find an $x$ such that $f \left(x\right) = y$.

There are a couple of ways to find the range of a function. For $f \left(x\right) = \text{-5} - 2 {\left(x + 3\right)}^{2}$, the easiest way is to see that $f$ is a certain type of function—a parabolic function. As such, there is no limitation on the $x$-values we can input (i.e. the domain is $\mathbb{R}$), but there is an easy-to-see limitation on the output (or $y$) values $f \left(x\right)$ can give.

Take a look at that squared bit—the ${\left(x + 3\right)}^{2}$. We know this can't be less than 0, because the square of a number is always positive (or 0, if we're squaring 0).

So ${\left(x + 3\right)}^{2}$ is no less than 0. Meaning, $2 {\left(x + 3\right)}^{2}$ is also no less than 0. Then $- 2 {\left(x + 3\right)}^{2}$ is no more than 0, and thus $\text{-} 5 - 2 {\left(x + 3\right)}^{2}$ is no more than -5.

In math:

$\textcolor{w h i t e}{f \left(x\right) = \text{-5} - 2} {\left(x + 3\right)}^{2} \ge 0$
$\textcolor{w h i t e}{f \left(x\right) = \text{-5} -} 2 {\left(x + 3\right)}^{2} \ge 0$
color(white)(f(x)="-5")-2(x+3)^2<=0" "(note the change to $\le$)
$f \left(x\right) = \text{-5"-2(x+3)^2<="-5}$

The end result: $f \left(x\right) \le \text{-5}$.
And so our range is "all numbers $y$ such that $y$ is at most -5", or in math:

$\left\{y | y \le \text{-5}\right\}$.

## Bonus:

The shortcut to finding the range of a parabolic equation $f \left(x\right) = a {\left(x - h\right)}^{2} + k$ is to use this:

Range of $f \left(x\right)$ is $\left\{\begin{matrix}\left\{y | y \ge k\right\} \text{ when "a>0 \\ {y|y<=k}" when } a < 0\end{matrix}\right.$

Just choose the correct option depending on the value of $a$. (In this example, $a$ was less than 0, so we would choose $\left\{y | y \le k\right\}$, and plug in our $k$-value of -5.)