Understanding the Absolute Value Function: Definition, Properties, and Applications in Mathematics

the absolute value function

The absolute value function is a mathematical function that returns the distance of a number from zero on a number line

The absolute value function is a mathematical function that returns the distance of a number from zero on a number line. It is denoted by the notation |x|, where x is the input value.

The absolute value function can be defined using a piecewise function as follows:

| x | = x if x ≥ 0
-x if x < 0 In simpler terms, the absolute value of a positive number is the number itself, while the absolute value of a negative number is its positive counterpart. For example, |3| = 3 since 3 is a positive number, and |(-5)| = 5 since -5 is a negative number. Here are some properties and applications of the absolute value function: 1. Non-negativity: The absolute value of any number is always non-negative. This means that |x| ≥ 0 for any value of x. 2. Symmetry: The absolute value function exhibits symmetry about the y-axis. This implies that |x| = |(-x)| for any value of x. 3. Distance: As mentioned earlier, the absolute value function represents the distance of a number from zero on a number line. For example, if you are driving 5 miles to the east and then 3 miles to the west, your total displacement from the starting point would be |5 - 3| = 2 miles. 4. Solving equations: The absolute value function is often used in solving equations. To solve an equation involving absolute value, one can consider two cases: when the expression inside the absolute value is non-negative and when it is negative. For instance, to solve |2x + 1| = 3, we consider both 2x + 1 = 3 and 2x + 1 = -3 separately to find the possible solutions. 5. Inequalities: The absolute value function can also be used to solve inequalities. When solving an inequality containing absolute value, it is necessary to consider both the positive and negative versions of the inequality. For example, to solve |x - 2| < 4, we consider both x - 2 < 4 and -(x - 2) < 4 separately to find the valid solution range. Understanding the properties and applications of the absolute value function can be helpful in various mathematical concepts such as algebra, calculus, and problem-solving.

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