A piecewise function uses one rule on one stretch of inputs and a different rule elsewhere.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Choose before computing
Decide which condition the input satisfies, then use only that rule.
The boundary belongs to one side
The conditions must not overlap, or one input would have two outputs — and it would not be a function.
Where they appear
Postage, tax bands, and phone tariffs are all piecewise. The rule changes at a threshold.
Different rules on different intervals
A piecewise function uses one formula on part of the domain and another elsewhere. Each piece comes with the interval where it applies, and together the intervals must cover the domain without overlapping.
Choose the piece first
To evaluate f(4), find which interval contains 4, then use that piece's formula. Using the wrong piece is the standard error, and it is avoided by checking the interval before touching the algebra.
Graphing piece by piece
Graph each formula only over its own interval, marking endpoints as open or closed to match the inequalities. The endpoints are where errors hide, so draw them deliberately rather than letting the pieces run together.
Where they occur
Tax brackets, postage rates, phone tariffs and parking charges are all piecewise. Any situation with a threshold that changes the rule is naturally piecewise, which makes them far more common in practice than in textbooks.
Step 2: Try It Yourself
Tap and try it out.
The slope is 2: for every 1 across, the line goes 2 up.
Step 3: Watch an Example
One step at a time.
Watch Priya Pick the Piece
f(x) = 2x for x < 3, and f(x) = x + 10 for x ≥ 3. Priya evaluates at 1 and at 5.
- Step 1
For x = 1: 1 is less than 3, so she uses 2x.
Step 4: Your Turn
Practice makes it stick.
Below the Threshold
Problem 1 of 2
f(x) = 3x for x < 4, else x + 20. What is f(2)?
Above It
Problem 2 of 2
Same function. What is f(10)?
Which Rule Applies
1 of 8
f(x) = 2x for x < 5, else 100. What is f(3)?
2 of 8
Same function. What is f(9)?
3 of 8
Same function. What is f(5)?
4 of 8
May two pieces both claim the same input? 1 for yes, 0 for no.
5 of 8
f(x) = x + 1 for x ≤ 0, else 4x. What is f(0)?
6 of 8
Same function. What is f(2)?
7 of 8
Order the steps for evaluating a piecewise function.
- 1Select that rule only
- 2Substitute into it
- 3Check which condition the input satisfies
8 of 8
Is a postage cost by weight band piecewise? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
f(x) = 5x for x < 2, else x + 50. What is f(1)?
What You Learned
- A piecewise function applies a different rule on each stretch of inputs.
- Decide which condition holds before computing anything.
- The conditions must not overlap, or it would not be a function at all.