To understand how the absolute value function [tex]\( f(x) = |x+8| \)[/tex] can be written as a piecewise function, we need to consider the definition of the absolute value function, which states that:
[tex]\[
|a| =
\begin{cases}
a & \text{if } a \geq 0 \\
-a & \text{if } a < 0
\end{cases}
\][/tex]
Applying this to our given function [tex]\( f(x) = |x+8| \)[/tex]:
1. Case 1: When [tex]\( x + 8 \geq 0 \)[/tex] (i.e., [tex]\( x \geq -8 \)[/tex]):
For this case, the expression inside the absolute value is non-negative, so we can remove the absolute value bars directly:
[tex]\[
f(x) = x + 8
\][/tex]
2. Case 2: When [tex]\( x + 8 < 0 \)[/tex] (i.e., [tex]\( x < -8 \)[/tex]):
For this case, the expression inside the absolute value is negative, so we must take the negative of the expression to make it positive:
[tex]\[
f(x) = -(x + 8) = -x - 8
\][/tex]
Putting these two cases together, we can express [tex]\( f(x) \)[/tex] as a piecewise function:
[tex]\[
f(x) =
\begin{cases}
x + 8 & \text{if } x \geq -8 \\
-x - 8 & \text{if } x < -8
\end{cases}
\][/tex]
Now we can match this piecewise function with the provided options:
- Option A: [tex]\( f(x)=\left\{\begin{array}{ll}x + 8, & x \geq -8 \\ -x - 8, & x < -8\end{array}\right. \)[/tex]
Option A is the correct answer, as it accurately represents the piecewise function for [tex]\( f(x) = |x+8| \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{\text{A}} \][/tex]
