To evaluate the piecewise function [tex]\( f(x) \)[/tex] at [tex]\( x = -1 \)[/tex], follow the steps below:
1. Identify which part of the piecewise function applies to [tex]\( x = -1 \)[/tex].
[tex]\[
f(x) = \left\{
\begin{array}{l}
x + 4, \quad x < -1 \\
x, \quad -1 \leq x \leq 3 \\
x + 1, \quad x > 3
\end{array}
\right.
\][/tex]
2. Determine the condition that [tex]\( x = -1 \)[/tex] meets:
- The first condition is [tex]\( x < -1 \)[/tex]: Here, [tex]\( x = -1 \)[/tex] does not satisfy this condition because [tex]\(-1\)[/tex] is not less than [tex]\(-1\)[/tex].
- The second condition is [tex]\( -1 \leq x \leq 3 \)[/tex]: Here, [tex]\( x = -1 \)[/tex] satisfies this condition because [tex]\(-1\)[/tex] is equal to [tex]\(-1\)[/tex], which is within the interval [tex]\([-1, 3]\)[/tex].
- The third condition is [tex]\( x > 3 \)[/tex]: Here, [tex]\( x = -1 \)[/tex] does not satisfy this condition because [tex]\(-1\)[/tex] is not greater than [tex]\( 3 \)[/tex].
3. Since [tex]\( x = -1 \)[/tex] satisfies the second condition [tex]\( -1 \leq x \leq 3 \)[/tex], the corresponding expression for [tex]\( f(x) \)[/tex] is [tex]\( x \)[/tex].
4. Substitute [tex]\( x = -1 \)[/tex] into the expression from the selected condition:
[tex]\[
f(-1) = -1
\][/tex]
Thus, [tex]\( f(-1) = -1 \)[/tex].
[tex]\[
\boxed{-1}
\][/tex]
