To find the value of the piecewise function [tex]\( f(x) \)[/tex] when [tex]\( x = 3 \)[/tex], let's analyze the conditions given in the definition of the piecewise function:
[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]
We are tasked with finding [tex]\( f(3) \)[/tex]. Let's match [tex]\( x = 3 \)[/tex] with the appropriate condition from the piecewise function definition:
1. For [tex]\( x \leq -1 \)[/tex]:
[tex]\[
f(x) = -x
\][/tex]
This condition does not apply because [tex]\( 3 \)[/tex] is not less than or equal to [tex]\(-1\)[/tex].
2. For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(x) = 1
\][/tex]
This condition does not apply because [tex]\( 3 \)[/tex] is not equal to [tex]\( 0 \)[/tex].
3. For [tex]\( x \geq 1 \)[/tex]:
[tex]\[
f(x) = x + 1
\][/tex]
This condition applies because [tex]\( 3 \)[/tex] is greater than or equal to [tex]\( 1 \)[/tex].
Since [tex]\( x = 3 \)[/tex] satisfies the third condition, we use the formula [tex]\( f(x) = x + 1 \)[/tex]:
[tex]\[
f(3) = 3 + 1 = 4
\][/tex]
Therefore, the value of [tex]\( f(3) \)[/tex] is [tex]\( 4 \)[/tex]. The corresponding answer is:
[tex]\[
\boxed{4}
\][/tex]
