Absolute Value Function
f(x) = a I x – h I + kDomain: ]-∞; ∞[Range: [k, ∞[
The absolute value function, denoted by |x|, is a mathematical function that takes a number as its input and returns the positive value of that number. For example, the absolute value of -3 is 3, and the absolute value of 6 is 6.
The absolute value function is defined as follows:
| x | = {
x, if x ≥ 0
-x, if x < 0
}
The graph of the absolute value function looks like a “V” shape. The vertex, or lowest point of the V, is (0,0) and it opens upwards as shown below:
|\ /
| \ /
| \ /
| \ /
| \/
| /\
| / \
| / \
| / \
|/ \
The absolute value function has many important properties in mathematics, including:
1. The absolute value function is always nonnegative: |x| ≥ 0 for all x.
2. The absolute value function is an odd function: |(-x)| = |x| for all x.
3. The absolute value function is continuous everywhere.
4. The absolute value function is differentiable everywhere except at x = 0.
5. The absolute value function can be used to represent the distance between two points on a number line. For example, the distance between points a and b is |a-b|.
6. The absolute value function satisfies the triangle inequality: |x + y| ≤ |x| + |y| for all x and y.
In summary, the absolute value function is an important mathematical concept that is used in a variety of applications, including geometry, calculus, and algebra.
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