The Absolute Value Function: Definition, Properties, Graph, and Applications

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.