Evaluate a Piecewise Function
Use x as the variable. Supported functions include sqrt, abs, sin, log, min, and max.
Example Data Table
These values use the included three-rule example. The endpoint at 5 belongs to Rule 3.
| x value | Selected rule | Formula used | Result |
|---|---|---|---|
| −2 | 1 | x^2 | 4 |
| 0 | 2 | 2*x+1 | 1 |
| 4 | 2 | 2*x+1 | 9 |
| 5 | 3 | sqrt(x)+3 | 5.2361 |
Formula Used
General piecewise rule:
f(x) = Ei(x), when x belongs to interval Ii.
The calculator finds every Ii containing x. It evaluates the first matching expression Ei(x). Open endpoints use < or >. Closed endpoints use ≤ or ≥.
How to Use This Calculator
- Select the number of rules in your function.
- Enter one expression for each active rule.
- Set lower and upper limits. Leave a limit blank for infinity.
- Select endpoint boxes for brackets. Leave them clear for parentheses.
- Enter the x value and choose your decimal precision.
- Press Calculate, then check the selected rule and final value.
Piecewise Function Evaluation Guide
Understanding Piecewise Functions
A piecewise function uses different formulas for different input ranges. Each range is a rule with its own interval. The function selects one rule after you enter an x value. This idea appears in algebra, science. A delivery charge can change after a weight limit. A tax amount can change after an income threshold. A graph can combine lines, curves, and flat sections. Piecewise notation keeps those changes organized. You supply a formula for each rule. Then you mark whether endpoints belong to the interval. The calculator finds the matching range and evaluates its expression. The result explains which rule was used. That feedback helps you find missing intervals and unclear boundaries early.
Reading Interval Boundaries
Interval limits decide which expression applies. A closed endpoint includes its stated number. An open endpoint leaves that number out. For example, x less than 2 excludes 2. The interval x greater than or equal to 2 includes 2. Enter a lower limit when a rule begins. Enter an upper limit when a rule ends. Leave a limit empty when the interval continues forever. A blank lower limit reaches left without end. A blank upper limit reaches right without end. Avoid accidental overlaps between rules. An overlap creates more than one possible match. This calculator reports an overlap and uses the first matching rule. Gaps can also leave an x value without an answer. Join neighboring ranges carefully when complete coverage is required.
Writing Valid Expressions
Expressions may contain x, numbers, parentheses, and standard operations. Use plus, minus, multiplication, division, and exponents. Write x^2 for x squared. Use parentheses to control grouping. For example, write (x+3)^2 when the full sum is squared. The calculator also accepts abs, sqrt, sin, cos, tan, log, ln, exp, floor, ceil, min, and max. Trigonometric functions use radians. Invalid operations show a clear message. Division cannot use a zero denominator. Square roots need nonnegative inputs. Logarithms need positive inputs. These limits come from standard mathematical domains. Check formulas before calculating. Then compare the selected expression with your expected rule. Choose more decimal places for detailed work. Exported results include the entered value, selected interval, and calculated output for later review.
Checking Your Final Result
Start by choosing the number of rules. Two or three rules fit many classroom problems. Add more rules when the model changes often. Type formulas without an equals sign. Each formula should describe y using x. Set every interval boundary carefully. Tick an endpoint box only when the boundary belongs to that rule. Enter the x value you want to test. Pick a decimal precision. Press Calculate to show the result above the form. Review the selected interval and formula. Confirm that the substitution is sensible. Reset restores the included sample values. A calculator supports reasoning but cannot replace graph checks. Sketching helps reveal jumps, gaps, and overlaps. Test boundary values separately. Exact boundaries produce reliable outputs for school, work, and practical models.
Frequently Asked Questions
What is a piecewise function?
It is a function with two or more formulas. Each formula applies only within a stated input interval. The intervals decide which formula produces the output.
What should I enter as a formula?
Enter an expression using x. Examples include 3*x-2, x^2, and sqrt(x)+1. Do not include an equals sign or function label.
How do I represent infinity?
Leave the lower limit blank for an interval that continues toward negative infinity. Leave the upper limit blank for an interval that continues toward positive infinity.
What do endpoint boxes mean?
A checked box includes that boundary value. It represents a square bracket. An unchecked box excludes that boundary value. It represents a parenthesis.
What happens when two intervals overlap?
The calculator reports the overlap and evaluates the first matching rule. Revise the boundaries when your function should have only one rule for each x value.
Why does the calculator show no matching rule?
Your x value may fall inside a gap, or an endpoint may be excluded. Review all limits and endpoint boxes. Add a rule when the domain needs complete coverage.
Can I use trigonometric functions?
Yes. Use sin, cos, tan, asin, acos, or atan. Trigonometric inputs are interpreted in radians.
Can I use constants such as pi?
Yes. Use pi for π and e for Euler’s number. For example, sin(pi/2) evaluates to 1.
Why is my square root or logarithm rejected?
Square roots require nonnegative values. Logarithms require positive values. These limits apply after the calculator substitutes your selected x value into the matching rule.
Does decimal precision change the calculation?
No. Precision changes how many digits are displayed and exported. The calculator keeps the evaluated numeric result before formatting it for your selected precision.
Why should I test endpoints separately?
Open and closed endpoints may select different formulas. Try every listed boundary and compare the selected rule. Exact boundaries help the calculator select the correct rule.