To determine the value of the piecewise function [tex]\( f(x) \)[/tex] at [tex]\( x = 3 \)[/tex], we need to identify which condition [tex]\( x = 3 \)[/tex] satisfies and then apply the corresponding function definition.
The piecewise function given is:
[tex]\[
f(x) =
\begin{cases}
-x, & \text{if } x \leq -1 \\
1, & \text{if } x = 0 \\
x + 1, & \text{if } x \geq 1
\end{cases}
\][/tex]
Let's evaluate each condition for [tex]\( x = 3 \)[/tex]:
1. [tex]\( x \leq -1 \)[/tex]:
- For this condition to be true, [tex]\( x \)[/tex] must be less than or equal to [tex]\(-1\)[/tex]. Since [tex]\( 3 \)[/tex] is not less than or equal to [tex]\(-1\)[/tex], this condition does not apply.
2. [tex]\( x = 0 \)[/tex]:
- For this condition to be true, [tex]\( x \)[/tex] must be exactly [tex]\( 0 \)[/tex]. Since [tex]\( 3 \)[/tex] is not equal to [tex]\( 0 \)[/tex], this condition does not apply.
3. [tex]\( x \geq 1 \)[/tex]:
- For this condition to be true, [tex]\( x \)[/tex] must be greater than or equal to [tex]\( 1 \)[/tex]. Since [tex]\( 3 \)[/tex] is greater than [tex]\( 1 \)[/tex], this condition applies.
Since [tex]\( 3 \)[/tex] meets the third condition [tex]\( x \geq 1 \)[/tex], we use the corresponding function rule [tex]\( x + 1 \)[/tex]:
[tex]\[ f(3) = 3 + 1 \][/tex]
Perform the calculation:
[tex]\[ f(3) = 4 \][/tex]
Therefore, the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 3 \)[/tex] is [tex]\( 4 \)[/tex].
The correct answer is [tex]\(\boxed{4}\)[/tex].
