To write the absolute value function [tex]\(f(x) = |x + 8|\)[/tex] as a piecewise function, we need to consider the definition of the absolute value function. The absolute value of a number [tex]\(a\)[/tex] is defined as follows:
[tex]\[
|a| =
\begin{cases}
a & \text{if } a \geq 0 \\
-a & \text{if } a < 0
\end{cases}
\][/tex]
In this case, our function is [tex]\( f(x) = |x + 8| \)[/tex]. We need to consider two cases based on the expression inside the absolute value sign [tex]\(x + 8\)[/tex]:
1. [tex]\( x + 8 \geq 0 \)[/tex] which simplifies to [tex]\( x \geq -8 \)[/tex]
2. [tex]\( x + 8 < 0 \)[/tex] which simplifies to [tex]\( x < -8 \)[/tex]
For [tex]\( x \geq -8 \)[/tex]:
[tex]\[
|x + 8| = x + 8
\][/tex]
So, [tex]\( f(x) = x + 8 \)[/tex].
For [tex]\( x < -8 \)[/tex]:
[tex]\[
|x + 8| = -(x + 8) = -x - 8
\][/tex]
So, [tex]\( f(x) = -x - 8 \)[/tex].
Putting these together, we have the piecewise function:
[tex]\[
f(x) =
\begin{cases}
x + 8 & \text{if } x \geq -8 \\
-x - 8 & \text{if } x < -8
\end{cases}
\][/tex]
Let’s check the options provided:
A. [tex]\( f(x) = \left\{\begin{array}{ll}x+8, & x \geq -8 \\ -x-8, & x<-8\end{array}\right. \)[/tex]
B. [tex]\( f(x) = \left\{\begin{array}{ll}x+8, & x \geq -8 \\ -x+8, & x<-8\end{array}\right. \)[/tex]
C. [tex]\( f(x) = \left\{\begin{array}{ll}x+8, & x \geq 8 \\ -x+8, & x<8\end{array}\right. \)[/tex]
D. [tex]\( f(x) = \left\{\begin{array}{ll}x+8, & x \geq 8 \\ -x-8, & x<8\end{array}\right. \)[/tex]
The correct piecewise function is:
A. [tex]\( f(x) = \left\{\begin{array}{ll}x+8, & x \geq -8 \\ -x-8, & x<-8\end{array}\right. \)[/tex]
So, the correct answer is:
[tex]\[
\boxed{A}
\][/tex]
