---
title: "Absolute Value as Piecewise Function"
slug: absolute-value-as-piecewise-function
type: evergreen
stime: 2025-01-15 @ 12:00AM
status: evergreen
certainty: certain
importance: 7
tags: [algebra, absolute-value]
---

Given an expression like $|x - a|$, its piecewise form is:

$$
|x - a| =
\begin{cases}
x - a, & \text{if } x \geq a \\
-(x - a), & \text{if } x < a
\end{cases}
$$

---

### Example

Rewrite $f(x) = |x - 2|$ as a piecewise function:

$$
f(x) =
\begin{cases}
x - 2, & \text{if } x \geq 2 \\
- (x - 2), & \text{if } x < 2
\end{cases}
=
\begin{cases}
x - 2, & x \geq 2 \\
- x + 2, & x < 2
\end{cases}
$$

This representation is useful when:

- Solving equations involving $|x - a|$
- Graphing the function
- Performing algebraic analysis

## Related

- [Definition of Absolute Value](definition-of-absolute-value.md)
- [Limit with Absolute Value - Piecewise Behavior](limit-with-absolute-value-piecewise-behavior.md)
- [Limit with Absolute Value - Jump Discontinuity](limit-with-absolute-value-jump-discontinuity.md)
- [Function Notation](function-notation.md)
- [Differential Calculus](differential-calculus.md)
- [When a Limit Exists](when-a-limit-exists.md)
- [Notation of a Limit](notation-of-a-limit.md)
- [Computing Basic Limits - Direct Substitution](computing-basic-limits-direct-substitution.md)
- [Computing Limits with Algebraic Simplification](computing-limits-with-algebraic-simplification.md)
- [Limit with Difference of Cubes](limit-with-difference-of-cubes.md)