To determine the value of [tex]\( f(-2) \)[/tex] for the given piecewise function, we follow these steps:
1. Identify the correct interval for the input value:
We need to find the value of [tex]\( f(x) \)[/tex] at [tex]\( x = -2 \)[/tex]. Look at the conditions of the piecewise function:
[tex]\[
f(x) = \begin{cases}
-x^2, & x < -2 \\
3, & -2 \leq x < 0 \\
x + 2, & x \geq 0
\end{cases}
\][/tex]
2. Determine which part of the piecewise function applies:
Since [tex]\( x = -2 \)[/tex], we check which interval includes [tex]\(-2\)[/tex]:
- For [tex]\( x < -2 \)[/tex], the function is [tex]\( f(x) = -x^2 \)[/tex]. This interval does not include [tex]\(-2\)[/tex].
- For [tex]\(-2 \leq x < 0 \)[/tex], the function is [tex]\( f(x) = 3 \)[/tex]. This interval includes [tex]\(-2\)[/tex].
3. Find the corresponding output value:
Since [tex]\(-2\)[/tex] is in the interval [tex]\(-2 \leq x < 0\)[/tex], we use the function defined in this interval, which is [tex]\( f(x) = 3 \)[/tex].
Therefore, [tex]\( f(-2) = 3 \)[/tex].
Among the given choices:
- [tex]$f(-2) = -6$[/tex]
- [tex]$f(-2) = -4$[/tex]
- [tex]$f(-2) = 0$[/tex]
- [tex]$f(-2) = 3$[/tex]
The correct answer is [tex]\(\boxed{3}\)[/tex].
