The equation [tex]\( f(x) = |x| \)[/tex] represents a function commonly known as the absolute value function. To understand why, let's break down the components and characteristics of this type of function:
1. Absolute Value Definition: The absolute value [tex]\( |x| \)[/tex] of a number [tex]\( x \)[/tex] is defined as:
- [tex]\( |x| = x \)[/tex] if [tex]\( x \geq 0 \)[/tex]
- [tex]\( |x| = -x \)[/tex] if [tex]\( x < 0 \)[/tex]
2. Graphical Representation: When you plot this function on a coordinate plane, it creates a "V" shape with its vertex at the origin (0, 0). This happens because the output is always non-negative, reflecting both positive and negative inputs onto the positive side of the y-axis.
3. Function Behavior: For any input value [tex]\( x \)[/tex], the function [tex]\( f(x) = |x| \)[/tex] returns the non-negative value of [tex]\( x \)[/tex]. For instance,
- If [tex]\( x = 3 \)[/tex], [tex]\( f(3) = |3| = 3 \)[/tex]
- If [tex]\( x = -3 \)[/tex], [tex]\( f(-3) = |-3| = 3 \)[/tex]
Given these explanations, option (A) is the correct answer. The equation [tex]\( f(x) = |x| \)[/tex] is known as the absolute value function.
