To express the absolute value function [tex]\( f(x) = |x - 2| \)[/tex] as a piecewise function, we need to consider the definition of the absolute value function.
The absolute value function [tex]\( |a| \)[/tex] is defined as:
[tex]\[
|a| =
\begin{cases}
a, & \text{if } a \geq 0 \\
-a, & \text{if } a < 0
\end{cases}
\][/tex]
Here, [tex]\( a = x - 2 \)[/tex]. So we need to determine in which cases [tex]\( x - 2 \)[/tex] is non-negative (greater than or equal to zero) and in which cases it is negative (less than zero).
1. When [tex]\( x - 2 \geq 0 \)[/tex]:
[tex]\[
x \geq 2
\][/tex]
In this case, [tex]\( |x - 2| = x - 2 \)[/tex].
2. When [tex]\( x - 2 < 0 \)[/tex]:
[tex]\[
x < 2
\][/tex]
In this case, [tex]\( |x - 2| = -(x - 2) = -x + 2 \)[/tex].
Putting this information together, the piecewise function for [tex]\( f(x) = |x - 2| \)[/tex] is:
[tex]\[
f(x) =
\begin{cases}
-x + 2, & \text{if } x < 2 \\
x - 2, & \text{if } x \geq 2
\end{cases}
\][/tex]
Therefore, the correct pieces that the function will include are:
[tex]\[
-x + 2 \text{ for } x < 2
\][/tex]
and
[tex]\[
x - 2 \text{ for } x \geq 2
\][/tex]
So, the correct answer is:
[tex]\[
-x + 2, \; x < 2
\][/tex]
and
[tex]\[
x - 2, \; x \geq 2
\][/tex]
