To find [tex]\( f(4) \)[/tex] for the piecewise function defined as:
[tex]\[
f(x) = \left\{
\begin{array}{ll}
x - 2 & \text{if } x < 3, \\
x - 1 & \text{if } x \geq 3
\end{array}
\right.
\][/tex]
we need to determine which part of the piecewise function to use by checking the condition that [tex]\( x \)[/tex] satisfies.
1. Start by noting the value of [tex]\( x \)[/tex] that we need to evaluate: [tex]\( x = 4 \)[/tex].
2. Check which condition [tex]\( x = 4 \)[/tex] satisfies:
- The first condition applies if [tex]\( x < 3 \)[/tex]. For [tex]\( x = 4 \)[/tex], this condition is not satisfied because 4 is not less than 3.
- The second condition applies if [tex]\( x \geq 3 \)[/tex]. For [tex]\( x = 4 \)[/tex], this condition is satisfied because 4 is greater than or equal to 3.
Since [tex]\( x = 4 \)[/tex] satisfies the second condition [tex]\( x \geq 3 \)[/tex], we use the corresponding expression for the second part of the piecewise function:
[tex]\[
f(x) = x - 1
\][/tex]
3. Substitute [tex]\( x = 4 \)[/tex] into the expression [tex]\( f(x) = x - 1 \)[/tex]:
[tex]\[
f(4) = 4 - 1
\][/tex]
4. Perform the subtraction:
[tex]\[
f(4) = 3
\][/tex]
Thus, the value of [tex]\( f(4) \)[/tex] is [tex]\( \boxed{3} \)[/tex].
