Absolute Value Function
The absolute value function, denoted as |x|, is a mathematical function that returns the non-negative value of a real number
The absolute value function, denoted as |x|, is a mathematical function that returns the non-negative value of a real number. It effectively measures the “distance” between the number and zero on the number line, regardless of the number’s sign.
For positive numbers, the absolute value function simply returns the number itself. For example, |4| = 4 and |7| = 7.
For negative numbers, the absolute value function negates the sign and returns the positive value. For example, |-3| = 3 and |-5| = 5.
In general, the absolute value function can be defined as follows:
For any real number x,
|x| =
x, if x ≥ 0,
-x, if x < 0.
This function can also be represented graphically as a "V" or "U" shape, with the vertex at the origin (0,0). The value of the function increases as we move away from zero in either direction, creating a reflection symmetry about the y-axis.
The absolute value function has several useful properties:
1. |x| ≥ 0 for any real number x.
2. |x| = 0 if and only if x = 0.
3. For any real numbers a and b, |a * b| = |a| * |b|.
4. For any real number x, |x + a| ≤ |x| + |a| (Triangle inequality).
The absolute value function is commonly used in mathematics, especially in solving equations and inequalities involving absolute values, distance calculations, and analyzing piecewise-defined functions.
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