To rewrite the absolute value function [tex]\( f(x) = |x + 3| \)[/tex] as a piecewise function, we need to determine how the function behaves for different intervals of [tex]\( x \)[/tex].
The absolute value function [tex]\( |x + 3| \)[/tex] can be split into two cases:
1. When the expression inside the absolute value, [tex]\( x + 3 \)[/tex], is non-negative (i.e., [tex]\( x + 3 \geq 0 \)[/tex]), then the absolute value does not change the expression.
- This happens when [tex]\( x \geq -3 \)[/tex].
- In this case, [tex]\( f(x) = x + 3 \)[/tex].
2. When the expression inside the absolute value, [tex]\( x + 3 \)[/tex], is negative (i.e., [tex]\( x + 3 < 0 \)[/tex]), then the absolute value changes the sign of the expression.
- This happens when [tex]\( x < -3 \)[/tex].
- In this case, [tex]\( f(x) = -(x + 3) = -x - 3 \)[/tex].
So, the piecewise function can be written as:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
x + 3 & \text{if } x \geq -3 \\
-x - 3 & \text{if } x < -3
\end{array}
\right.
\][/tex]
Placing the appropriate pieces into the given format, we get:
[tex]\[
\begin{array}{c}
x < -3 \quad x \geq -3 \quad -x - 3 \quad x + 3 \quad x \geq 3 \quad x - 3 \quad x < 3 \\
f(x) = \left\{
\begin{array}{ll}
x + 3 & \text{if } x \geq -3 \\
-x - 3 & \text{if } x < -3
\end{array}
\right.
\end{array}
\][/tex]
So the correct tiles to use are:
- [tex]\( x \geq -3 \)[/tex]
- [tex]\( x < -3 \)[/tex]
- [tex]\( x + 3 \)[/tex]
- [tex]\( -x - 3 \)[/tex]
