To determine the correct piece of the piecewise function for [tex]\( f(x) = |x+3| \)[/tex], we need to express the absolute value function without the absolute value sign.
We can write the absolute value function as a piecewise function based on the definition of absolute value:
[tex]\[
|x+3| =
\begin{cases}
x + 3 & \text{if } x + 3 \geq 0 \\
-(x + 3) & \text{if } x + 3 < 0
\end{cases}
\][/tex]
We simplify the conditions:
1. [tex]\(x + 3 \geq 0 \)[/tex] simplifies to [tex]\( x \geq -3 \)[/tex].
2. [tex]\(x + 3 < 0\)[/tex] simplifies to [tex]\( x < -3 \)[/tex].
This gives us the piecewise function:
[tex]\[
f(x) =
\begin{cases}
x + 3 & \text{if } x \geq -3 \\
-(x + 3) & \text{if } x < -3
\end{cases}
\][/tex]
Now, let's identify which of the provided choices matches these pieces:
A. [tex]\(x + 3, x \geq -3\)[/tex]
B. [tex]\(-x + 3, x < -3\)[/tex]
C. [tex]\(-x - 3, x < 3\)[/tex]
D. [tex]\(x + 3, x \geq 3\)[/tex]
By comparing, we see:
- Choice A: [tex]\(x + 3, x \geq -3\)[/tex] matches our first condition.
- Choice B: [tex]\(-x + 3, x < -3\)[/tex] does not match any of our conditions.
- Choice C: [tex]\(-x - 3, x < 3\)[/tex] does not match any of our conditions.
- Choice D: [tex]\(x + 3, x \geq 3\)[/tex] does not match any of our conditions.
Therefore, the correct piece that is included in the piecewise function for [tex]\( f(x)=|x+3| \)[/tex] is:
\[
\boxed{A. \, x + 3, x \geq -3}
