To rewrite the absolute value function [tex]\( f(x) = |x + 3| \)[/tex] as a piecewise function, we need to consider the behavior of the function based on the value of [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Identify the expression inside the absolute value: The function given is [tex]\( f(x) = |x + 3| \)[/tex].
2. Determine when the expression inside the absolute value is non-negative or negative:
- [tex]\( x + 3 \geq 0 \)[/tex]
- [tex]\( x + 3 < 0 \)[/tex]
3. Solve these inequalities to find the critical value:
- For [tex]\( x + 3 \ge 0 \)[/tex], simplify to [tex]\( x \ge -3 \)[/tex]
- For [tex]\( x + 3 < 0 \)[/tex], simplify to [tex]\( x < -3 \)[/tex]
4. Write the piecewise function based on these intervals:
- When [tex]\( x \ge -3 \)[/tex], the expression [tex]\( x + 3 \)[/tex] is non-negative, so [tex]\( |x + 3| = x + 3 \)[/tex]
- When [tex]\( x < -3 \)[/tex], the expression [tex]\( x + 3 \)[/tex] is negative, so [tex]\( |x + 3| = -(x + 3) = -x - 3 \)[/tex]
5. Combine these pieces to form the piecewise function:
[tex]\[
f(x) =
\begin{cases}
x + 3, & \text{if } x \ge -3 \\
-x - 3, & \text{if } x < -3
\end{cases}
\][/tex]
So, filling in the blanks in the equation, we get:
[tex]\[
\begin{array}{l}
f(x)=\left\{\begin{array}{l}
x + 3, \quad \text{if } x \geq -3 \\
-x - 3, \quad \text{if } x < -3
\end{array}\right.
\end{array}
\][/tex]
Hence, the absolute value function [tex]\( f(x) = |x + 3| \)[/tex] can be rewritten as the piecewise function as shown above.
