📐 Piecewise-Defined Functions
Big Idea
A piecewise-defined function is a function defined by different formulas on different parts of its domain. Each formula applies only to a specific interval of $x$ values. Think of it as a set of rules — which rule you use depends on the input!
🚕 Why Do We Need Piecewise Functions?
Imagine a taxi that charges:
- $3 flat fee for the first kilometer
- $1.50 per kilometer for every kilometer after that
The pricing rule changes depending on the distance! For a 0.5 km ride, you pay $3. For a 4 km ride, you pay $3 + $1.50 · 3 = $7.50. This is a piecewise function in real life.
Other real-life examples include:
🅿️ Parking Fees
Free for 1st hour, $2/hr after
Free for 1st hour, $2/hr after
⚡ Electric Bills
Different rates per usage level
Different rates per usage level
📝 Grade Scales
A for 90–100, B for 80–89, etc.
A for 90–100, B for 80–89, etc.
📦 Shipping Cost
Flat rate up to 5 kg, more above
Flat rate up to 5 kg, more above
📖 How to Read Piecewise Notation
A piecewise function is written with a large curly brace on the left. Each line shows a formula and the interval (condition) where it applies.
Example Notation:
$$f(x) = \begin{cases} 2x + 1, & \text{if } x < 0 \\ x^2, & \text{if } x \geq 0 \end{cases}$$
Piece 1: Use $2x + 1$ when $x$ is negative (less than 0)
Piece 2: Use $x^2$ when $x$ is zero or positive ($x \geq 0$)
⚠️ Key Rule: Every $x$ value belongs to exactly one interval. The boundary point (like $x = 0$ above) belongs to the piece that includes the equals sign ($\leq$ or $\geq$), not both.