Piecewise Functions
Defining functions with multiple rules
A piecewise function uses different rules for different parts of its domain. Instead of one formula for every input, the function tells you which rule to use based on the value of x. This makes piecewise functions useful for real-world situations such as shipping prices, tax brackets, phone plans, and motion graphs.
What Is a Piecewise Function?
A piecewise function is a function made from two or more formulas. Each formula applies only when its condition is true. The conditions divide the domain into separate intervals.
Basic Structure
How to Read the Rules
When evaluating a piecewise function, do not choose a formula by guessing. First look at the input value, then find the condition that includes it. Use only that matching rule.
Example Function
The boundary value x = 1 belongs to the second rule because its condition is x ≥ 1.
Step-by-Step Evaluation
Find f(−3)
Because −3 is less than 1, use the rule x + 2:
f(−3) = −3 + 2 = −1
Find f(4)
Because 4 is greater than or equal to 1, use the rule 3x − 1:
f(4) = 3(4) − 1 = 12 − 1 = 11
Check the Boundary: Find f(1)
Since 1 ≥ 1, use the second rule:
f(1) = 3(1) − 1 = 2
Memory Tip
The condition is the gatekeeper. Read the input first, match it to a condition, and only then substitute it into the correct formula.
Graphing a Piecewise Function
To graph a piecewise function, graph each formula only on the interval assigned to it. The endpoint symbols are important: an open circle means the endpoint is not included, while a closed circle means the endpoint is included.
| Symbol | Meaning | Graph point |
|---|---|---|
| x < a | Less than a | Open endpoint at a |
| x ≤ a | Less than or equal to a | Closed endpoint at a |
| x > a | Greater than a | Open endpoint at a |
| x ≥ a | Greater than or equal to a | Closed endpoint at a |
On a phone, swipe left or right inside the table if needed.
Worked Graphing Example
Consider the function with two rules: f(x) = x + 1 when x < 2, and f(x) = 4 when x ≥ 2.
First piece: Graph y = x + 1 only to the left of x = 2. At x = 2, this rule would give y = 3, but x = 2 is not included, so use an open circle at (2, 3).
Second piece: Graph y = 4 starting at x = 2 and moving right. Because x = 2 is included, use a closed circle at (2, 4).
Continuity at a Boundary
A piecewise function is continuous at a boundary when the pieces meet without a jump or gap. To check a boundary, compare the value from the left rule, the value from the right rule, and the actual function value at the boundary.
Continuous Example
The rules x + 1 for x < 2 and 3x − 3 for x ≥ 2 both give 3 when x = 2. The pieces meet at the same height, so the function can be continuous there.
Real-World Example: Delivery Pricing
A delivery company might charge one price for packages weighing up to 5 pounds and a higher price for heavier packages. A piecewise function can describe this pricing rule clearly.
The rule changes when the package weighs more than 5 pounds.
Common Mistakes to Avoid
Using every rule: Use only the rule whose condition matches the input.
Ignoring equality: x < 2 does not include 2, but x ≤ 2 does.
Mixing endpoint circles: Open circles exclude a point; closed circles include it.
Graphing outside the interval: Draw each formula only where its condition allows it.
Practice Problems
1. For f(x) = x + 4 when x < 0 and f(x) = 2x − 1 when x ≥ 0, find f(−3).
2. Using the same function, find f(5).
3. Is the endpoint included for the condition x > 7?
4. What type of circle represents an included endpoint?
Tap to reveal the answers
1. f(−3) = −3 + 4 = 1. 2. f(5) = 2(5) − 1 = 9. 3. No, because x > 7 excludes 7. 4. A closed circle.
Final Takeaway
A piecewise function is a collection of rules controlled by conditions. To evaluate or graph one, identify the correct interval first, apply only its formula, and pay close attention to open and closed endpoints.
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