| x | Matched Piece | Condition | Expression | Value | Status |
|---|
This example shows a three-part piecewise rule and sample results.
| Piece | Expression | Condition | Sample x | Expected Result |
|---|---|---|---|---|
| Left curve | x^2 + 2*x + 1 | x < 0 | -2 | 1 |
| Middle line | 3*x - 1 | x >= 0 && x < 5 | 2 | 5 |
| Right root | sqrt(x) + 10 | x >= 5 | 9 | 13 |
A piecewise function uses different expressions over different intervals.
f(x) = {
expression 1, if condition 1 is true
expression 2, if condition 2 is true
expression 3, if condition 3 is true
}
The calculator tests each condition in order. When a condition is true, its expression is evaluated with the chosen value of x.
If no condition is true, the answer is marked as unmatched unless another unmatched option is selected.
Endpoint checks are important. The condition x < 5 excludes 5.
The condition x <= 5 includes 5.
This difference can change the final answer.
- Enter one or more x values. You can also use range notation.
- Add every piece of your function with a label, expression, and condition.
- Place narrow or special rules before wider rules.
- Select precision, angle mode, unit label, and unmatched handling.
- Click the evaluate button. The result appears below the header and above the form.
- Review matched pieces, steps, and endpoint audit results.
- Download CSV for spreadsheets or PDF for printable reports.
Piecewise Function Evaluation Guide
Piecewise functions model rules that change across intervals. One formula may work before a cutoff. Another formula may work after it. This pattern appears in tax tables, shipping rates, signal systems, grading rules, and pricing plans. A normal calculator often hides which rule was used. This tool shows the matched piece and the final value.
Why Intervals Matter
Each piece has a condition. The condition tells when its expression is active. For example, x < 0 can use one expression. The range x >= 0 and x < 5 can use another. The endpoint is important. A closed endpoint includes the number. An open endpoint excludes it. Small endpoint mistakes can change the answer. This calculator helps expose those mistakes.
Advanced Evaluation Features
You can enter several x values at once. The calculator checks each value against every rule in order. The first true rule is used. You can add labels for better reports. You can set decimal precision. You can choose radians or degrees for trigonometric expressions. The endpoint scan also reviews boundary numbers found inside conditions.
Where It Helps
Teachers can prepare examples quickly. Students can verify homework steps. Engineers can test control rules. Business users can model tiered costs. Analysts can convert policy tables into numeric results. The export buttons make sharing easier. CSV works well with spreadsheets. PDF works well for printed notes.
How To Read The Result
The output table lists the input value, matched label, condition, expression, and answer. The steps explain the decision path. If no condition matches, the row is marked unmatched. That means the piecewise definition has a gap. Add an otherwise rule when every value needs an answer. Keep conditions ordered from most specific to most general. That avoids accidental matches.
Best Practices
Use simple expressions first. Test boundary values next. Then test values far inside each interval. Review each condition for overlap. Overlap is not always wrong, but order matters. Give every piece a clear label. Keep exports with your notes. They provide a useful audit trail. Clear documentation also reduces disputes. A visible rule path makes each answer easier to check, explain, revise, and approve later confidently.
1. What is a piecewise function?
A piecewise function uses different formulas for different intervals or conditions. The value of x decides which formula is active. This calculator checks the condition and then evaluates the matching expression.
2. Why does rule order matter?
The calculator reads rules from top to bottom. If two conditions overlap, the first matching rule is used. Put special cases first and broad conditions later.
3. Can I enter many x values?
Yes. Enter values separated by commas, spaces, or new lines. You can also enter ranges like 0:1:10 to generate values from 0 to 10.
4. What does otherwise mean?
Otherwise acts as a default condition. It matches any value not matched by earlier rules. It is useful when the function should always return an answer.
5. How are endpoints checked?
The endpoint scan finds numbers inside conditions. It tests values just below, equal to, and just above those numbers. This helps find open and closed interval mistakes.
6. Which math functions are supported?
The calculator supports common functions such as sin, cos, tan, sqrt, abs, log, ln, exp, floor, ceil, round, min, max, and pow.
7. Why do I get an unmatched result?
An unmatched result means no condition was true for that x value. Add another interval, fix a boundary, or include an otherwise rule.
8. What do the export buttons do?
The CSV button downloads a spreadsheet friendly file. The PDF button creates a printable report with results, matched pieces, and evaluation steps.