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 and how it affects the expression based on the input value of [tex]\(x\)[/tex].
The absolute value function [tex]\( |x+3| \)[/tex] has different expressions based on whether [tex]\( x+3 \)[/tex] is non-negative or negative:
1. When [tex]\( x + 3 \)[/tex] is non-negative (i.e., [tex]\( x + 3 \geq 0 \)[/tex]), the absolute value function [tex]\( |x+3| \)[/tex] is simply [tex]\( x + 3 \)[/tex].
2. When [tex]\( x + 3 \)[/tex] is negative (i.e., [tex]\( x + 3 < 0 \)[/tex]), the absolute value function [tex]\( |x+3| \)[/tex] is [tex]\( -(x + 3) \)[/tex], which simplifies to [tex]\( -x - 3 \)[/tex].
To determine the conditions under which each expression applies:
- [tex]\( x + 3 \geq 0 \)[/tex] simplifies to [tex]\( x \geq -3 \)[/tex]
- [tex]\( x + 3 < 0 \)[/tex] simplifies to [tex]\( x < -3 \)[/tex]
Given these conditions, we can write the piecewise function as follows:
[tex]\[
f(x) = \left\{\begin{array}{ll}
x + 3, & \text{if } x \geq -3 \\
-x - 3, & \text{if } x < -3 \\
\end{array}\right.
\][/tex]
