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 function. The absolute value of a number is defined as follows:
[tex]\[
|a| =
\begin{cases}
a & \text{if } a \geq 0 \\
-a & \text{if } a < 0
\end{cases}
\][/tex]
Applying this to our function [tex]\( f(x) = |x + 3| \)[/tex]:
1. Case 1: When the expression inside the absolute value is non-negative ([tex]\( x + 3 \geq 0 \)[/tex]), the function [tex]\( f(x) \)[/tex] simplifies to:
[tex]\[
f(x) = x + 3
\][/tex]
This happens when:
[tex]\[
x + 3 \geq 0 \implies x \geq -3
\][/tex]
2. Case 2: When the expression inside the absolute value is negative ([tex]\( x + 3 < 0 \)[/tex]), the function [tex]\( f(x) \)[/tex] becomes the negative of the expression:
[tex]\[
f(x) = -(x + 3)
\][/tex]
This occurs when:
[tex]\[
x + 3 < 0 \implies x < -3
\][/tex]
Combining these cases, we can write the function [tex]\( f(x) = |x + 3| \)[/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 absolute value function [tex]\( f(x) = |x + 3| \)[/tex] is written as the piecewise function:
[tex]\[
f(x) =
\begin{cases}
x + 3 & \text{when } x \geq -3 \\
-(x + 3) & \text{when } x < -3
\end{cases}
\][/tex]
