To identify the function family to which [tex]\( f \)[/tex] belongs, let's carefully examine the form of the given function:
[tex]\[
f(x) = |x - 2|
\][/tex]
This function [tex]\( f \)[/tex] is defined using the absolute value operator, which is denoted by [tex]\( | \cdot | \)[/tex]. The absolute value function is characterized by its output, which is always non-negative regardless of whether the input within the absolute value signs is positive or negative.
Here are the key points to consider:
1. Absolute Value Definition: The absolute value function transforms a number into its non-negative form. Mathematically, for any real number [tex]\( y \)[/tex]:
[tex]\[
|y| =
\begin{cases}
y & \text{if } y \geq 0 \\
-y & \text{if } y < 0
\end{cases}
\][/tex]
2. Absolute Value in [tex]\( f \)[/tex]: The function [tex]\( f(x) = |x - 2| \)[/tex] uses this operation on the expression [tex]\( x - 2 \)[/tex]. This means for any value of [tex]\( x \)[/tex]:
[tex]\[
f(x) =
\begin{cases}
x - 2 & \text{if } x \geq 2 \\
2 - x & \text{if } x < 2
\end{cases}
\][/tex]
The behavior described above is typical of the absolute value function.
3. Typical Shape and Graph: The graph of an absolute value function typically has a characteristic "V" shape. For [tex]\( f(x) = |x - 2| \)[/tex], the vertex of the "V" is at the point [tex]\((2, 0)\)[/tex] on the Cartesian plane, and the arms of the "V" open upwards.
Given our observations and the key characteristics of the function, it's clear that [tex]\( f(x) = |x-2| \)[/tex] belongs to the Absolute Value function family.
Thus, the function [tex]\( f \)[/tex] is best categorized under the fourth option:
- [tex]\( \boxed{\text{Absolute Value}} \)[/tex]
