Piecewise functions use different rules for different inputs. They appear regularly on the ACT and are straightforward once you learn the process.
What Is a Piecewise Function?
A piecewise function is defined by multiple formulas, each applying to a specific interval:
This means: use when is negative, and when is zero or positive.
The 3-Step Process
Step 1: Check which condition your input satisfies.
Step 2: Use the corresponding formula.
Step 3: Evaluate.
Example: Find using the function above.
Step 1: , so use the first piece ().
Step 2: .
Example: Find .
Step 1: , so use the second piece ().
Step 2: .
Graphing Piecewise Functions
Graph each piece on its own interval. Use:
- Closed circles () for or (value IS included)
- Open circles () for or (value is NOT included)
At the boundary point, only ONE piece should have a closed circle (the one with or ).
The Step Function (Greatest Integer)
The greatest integer function (also written ) gives the largest integer less than or equal to :
- (NOT !)
The negative case trips people up. because is the largest integer that is .
Real-World Applications
Piecewise functions model many real scenarios:
- Tax brackets (different rates for different income ranges)
- Shipping costs (flat rate up to a weight, then per-pound after)
- Parking fees (first hour free, then hourly rate)
The ACT loves these application problems.
Common Mistakes to Avoid
- Using the wrong piece: Always check the condition FIRST. The most common error is evaluating with the wrong formula.
- Boundary confusion: At with conditions and , use the second piece (because is true).
- Floor function with negatives: , not . Go MORE negative, not less.
ACT Pro Tip
Piecewise function questions on the ACT are almost always about evaluation: given , find . Spend 3 seconds identifying which piece to use, then it becomes a standard algebra problem. Do not let the notation intimidate you.
Practice piecewise functions with our Piecewise Functions lesson — ACT-exclusive content.