Absolute Value Function
The absolute value function is a mathematical function that gives the distance of a number from zero on the number line
The absolute value function is a mathematical function that gives the distance of a number from zero on the number line. It is denoted by two vertical bars around the number. For example, the absolute value of -5 is written as | -5 | and equals 5.
Mathematically, the absolute value of a number x is defined as:
| x | = {
x, if x ≥ 0
– x, if x < 0
}
The absolute value function always produces a non-negative value because it simply measures the distance between a number and zero, ignoring its direction or sign.
Graphically, the absolute value function represents a V-shaped curve or "V" curve. The vertex of the V is at the origin (0, 0) and the arms extend equally in the positive and negative direction along the number line.
Some properties and characteristics of the absolute value function include:
1. Symmetry: The absolute value function is symmetric about the y-axis, meaning that the graph remains the same when reflected across the y-axis.
2. Range: The range of the absolute value function is all non-negative real numbers, as the absolute value always produces a non-negative value.
3. Transformation: The absolute value function can be transformed by shifting it horizontally or vertically. For example, adding a constant to the input of the function shifts the graph horizontally, while adding a constant to the output shifts the graph vertically.
4. Absolute value inequalities: The absolute value function is frequently used in solving absolute value inequalities. These types of inequalities involve an absolute value expression with a variable, and the goal is to determine the range of values that satisfy the inequality.
Overall, the absolute value function is a fundamental concept in mathematics and has various applications in algebra, calculus, and real-life situations involving distance or magnitude.
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