To rewrite the absolute value function [tex]\( f(x) = |x + 3| \)[/tex] as a piecewise function, we need to consider the definition of the absolute value. The absolute value function [tex]\( |x + 3| \)[/tex] can be rewritten depending on whether the expression inside the absolute value, [tex]\( x + 3 \)[/tex], is non-negative (i.e., zero or positive) or negative.
The function [tex]\( f(x) = |x + 3| \)[/tex] can be split into two cases:
1. When [tex]\( x + 3 \geq 0 \)[/tex], or equivalently [tex]\( x \geq -3 \)[/tex]:
In this case, the absolute value function becomes [tex]\( f(x) = x + 3 \)[/tex].
2. When [tex]\( x + 3 < 0 \)[/tex], or equivalently [tex]\( x < -3 \)[/tex]:
In this case, the absolute value function becomes [tex]\( f(x) = -(x + 3) \)[/tex], which simplifies to [tex]\( f(x) = -x - 3 \)[/tex].
Putting these cases together, we can rewrite [tex]\( f(x) \)[/tex] as a piecewise function:
[tex]\[
f(x) = \begin{cases}
x + 3 & \text{if } x \geq -3 \\
-x - 3 & \text{if } x < -3
\end{cases}
\][/tex]
Thus, the correct arrangement of the tiles and final function is:
[tex]\[
f(x) = \begin{cases}
x + 3, & \text{if } x \geq -3 \\
-x - 3, & \text{if } x < -3
\end{cases}
\][/tex]
This matches the solution given.
