Piecewise Functions
Learn how to work with functions defined by different rules over different parts of their domain. Master the skills needed to understand, graph, and analyze piecewise functions.
Learning Objectives
- Understand piecewise function notation
- Evaluate piecewise functions
- Graph piecewise functions
- Determine continuity
Introduction
Piecewise functions are unique mathematical functions that are defined by different rules or expressions over different parts of their domain. These functions are particularly useful when we need to describe situations where behavior changes at certain points or within certain intervals.
Real-world applications of piecewise functions include modeling situations like tax brackets, postal rates, utility bills, and physical phenomena that exhibit different behaviors under different conditions. Understanding how to work with piecewise functions is essential for both mathematical analysis and practical problem-solving.
Key Concepts
Function Definition
A piecewise function uses different formulas for different parts of its domain. Each piece is defined by both a function rule and a domain interval.
Domain Intervals
The domain is split into intervals using inequalities. These intervals must not overlap and should cover the entire intended domain of the function.
Continuity
A piecewise function is continuous if the pieces connect smoothly at their boundary points. This requires the function values to be equal at these points.
Common Examples
Important examples include the absolute value function |x|, step functions, and functions with restricted domains like √x and 1/x.
Interactive Examples
Absolute Value as a Piecewise Function
Learn how the absolute value function can be written as a piecewise function
x < 0 ? -x : x
The absolute value function |x|
Step Function
Explore a simple step function
x < 0 ? -1 : 1
Start with a basic step function
Practice Problems
Basic Piecewise Function
What is the value of f(2) for the function f(x) = { x² if x ≥ 0, -x if x < 0 }?
Domain and Range
What is the domain of f(x) = { √x if x ≥ 0, x - 1 if x < 0 }?
Continuity
Is the function f(x) = { x + 1 if x ≤ 2, 2x - 1 if x > 2 } continuous at x = 2?