Piecewise Function Calculator

Piecewise Function Calculator: Graph and Evaluate Step Functions

What Is a Piecewise Function?

A piecewise function is defined by different formulas for different intervals of the input. Example:

f(x) = {
    -x       if x < 0
    x²       if 0 ≤ x < 2
    log(x)   if x ≥ 2
}

Each “piece” has:

• A condition (when it applies)
• An expression (the formula to use)

How It Works (The Math)

The calculator evaluates functions using:

1. Input Parsing:

   • Conditions: x < 0, x >= 5, etc.
   • Expressions: 2x+3, sin(x), sqrt(x^2+1)

2. Evaluation Logic:

For each x value:
Check conditions top-to-bottom
Use first matching expression

3. Graphing:
   • Plots points at specified intervals
   • Colors each piece differently

How to Use the Calculator

1. Define Pieces:

   • Click ”+ Add Another Piece” for multiple segments
   • Set conditions (e.g., x > 1)
   • Enter expressions (e.g., 3x-2)

2. Evaluate:

   • Enter specific x values to calculate f(x)
   • Adjust graph range (X Min/Max)

3. View Results:

   • Interactive graph with zoom
   • Table of calculated values
   • Color-coded by piece

Key Features

✔ Multiple function pieces
✔ Supports all math.js operations
✔ Adjustable graphing precision
✔ Mobile-friendly interface
✔ No installation required

Frequently Asked Questions (FAQs)

Q: What math operations are supported?
A: All standard operations: +, -, *, /, ^, sin(), log(), etc. (via math.js)

Q: Can I graph discontinuous functions?
A: Yes! The calculator handles jumps and undefined points.

Q: Why is my graph not smooth?
A: Decrease the “Step Size” for higher precision (will be slower).

Q: How many pieces can I add?
A: Technically unlimited, but performance may degrade beyond 10 pieces.

Terminology Explained

• Piece: A function segment with its own rule
• Condition: The “if” statement (e.g., x < 0)
• Expression: The formula to use when condition is true
• Discontinuity: Where the function “jumps” between pieces
• Step Size: Distance between calculated points

Technical Details

Math Engine:
Powered by math.js (supports 200+ functions/constants)

Graphing Library:
Chart.js with custom piecewise adapter

Calculation Method:

1. Tests conditions sequentially
2. Evaluates first matching expression
3. Plots points at user-defined intervals

Limitations:

• Conditions must use x as the variable
• Complex numbers not fully supported
• Very small step sizes may crash browsers

Important Notes

1. Condition Order Matters - First matching condition wins
2. Undefined Areas - Will show gaps in graph
3. Performance - Large ranges + small steps = slow
4. Mobile Usage - Pinch-to-zoom supported
5. Keyboard Shortcuts - Tab between fields

Example Uses

• Education: Visualize step functions
• Engineering: Model system responses
• Economics: Create tiered pricing models
• Programming: Test conditional logic

Why It’s Useful

1. Saves Time - No manual point calculations
2. Reduces Errors - Automatic condition checking
3. Enhances Learning - Immediate visual feedback
4. Professional Results - Exportable graphs