To find [tex]\( f(3) \)[/tex] for the given piece-wise function, let's follow these steps in detail:
The piece-wise function [tex]\( f(x) \)[/tex] is defined as follows:
[tex]\[
f(x) =
\begin{cases}
x - 2 & \text{if } x < 3 \\
x - 1 & \text{if } x \geq 3
\end{cases}
\][/tex]
We need to evaluate this function at [tex]\( x = 3 \)[/tex].
1. Identify which part of the piece-wise function to use:
- The function [tex]\( f(x) = x - 2 \)[/tex] is applicable when [tex]\( x < 3 \)[/tex].
Since [tex]\( x = 3 \)[/tex] is not less than 3, this part does not apply.
- The function [tex]\( f(x) = x - 1 \)[/tex] is applicable when [tex]\( x \geq 3 \)[/tex].
Since [tex]\( x = 3 \)[/tex] is equal to 3, this part does apply.
2. Use the correct part of the function to calculate [tex]\( f(3) \)[/tex]:
- Since [tex]\( x = 3 \)[/tex] falls under the condition [tex]\( x \geq 3 \)[/tex], we use [tex]\( f(x) = x - 1 \)[/tex].
3. Substitute [tex]\( x = 3 \)[/tex] in the appropriate part of the function:
[tex]\[
f(3) = 3 - 1
\][/tex]
4. Perform the calculation:
[tex]\[
f(3) = 2
\][/tex]
Therefore, the value of [tex]\( f(3) \)[/tex] is:
[tex]\[
f(3) = 2
\][/tex]
