1. What Is the Absolute Value Function?
The absolute value function returns the magnitude of a number without considering its sign. It is defined as:
|x| =
x if x ≥ 0
-x if x < 0
Therefore, the output of the absolute value function is always non‑negative.
2. Properties of the Absolute Value Function
- The output is always ≥ 0
- The graph has a V‑shape
- There is a corner (non‑differentiable point) at x = 0
- The function is even:
|-x| = |x| - Represents the distance of a number from zero
3. Graph of the Absolute Value Function
The graph of f(x) = |x| is a V‑shaped graph with its vertex at the origin.
Graph characteristics:
- For x ≥ 0: a line with slope 1
- For x < 0: a line with slope -1
- Symmetric about the y‑axis
4. Domain and Range
- Domain: all real numbers (ℝ)
- Range: all non‑negative real numbers
[0, ∞)
5. Derivative of the Absolute Value Function
The absolute value function is differentiable everywhere except at x = 0:
f(x) = |x|
f'(x) = 1 if x > 0
f'(x) = -1 if x < 0
No derivative at x = 0
6. Solving Equations with Absolute Value
To solve absolute value equations, consider both the positive and negative cases.
Example:
|x - 3| = 5
Case 1: x - 3 = 5 → x = 8
Case 2: x - 3 = -5 → x = -2
7. Solving Absolute Value Inequalities
Example 1: Less than
|x - 2| < 3
This becomes a compound inequality:
-3 < x - 2 < 3
→ -1 < x < 5
Example 2: Greater than
|x| > 4
Two separate intervals:
x > 4 or x < -4
8. Applications of the Absolute Value Function
- Measuring distance on the number line
- Modeling error and deviation
- Analyzing V‑shaped graphs
- Solving equations and inequalities
- Applications in geometry, physics, and economics
9. More Examples
Example 1:
f(x) = |x + 1|
Vertex at x = -1
Example 2:
f(x) = |2x - 4|
Slopes: 2 and -2
Vertex at x = 2
10. Conclusion
The absolute value function is a fundamental mathematical function that returns the magnitude of a number regardless of its sign. Its V‑shaped graph, non‑negative range, and wide applications make it essential in algebra, calculus, and real‑world modeling.