Piecewise Function Grapher
body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f4f7f6;
color: #333;
line-height: 1.6;
margin: 0;
padding: 20px;
}
.loan-calc-container {
max-width: 900px;
margin: 30px auto;
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
display: flex;
flex-wrap: wrap;
gap: 30px;
}
h1, h2, h3 {
color: #004a99;
text-align: center;
margin-bottom: 20px;
}
.input-section, .output-section, .article-section {
flex: 1;
min-width: 300px;
}
.input-section h2 {
margin-top: 0;
text-align: left;
}
.input-group {
margin-bottom: 18px;
display: flex;
align-items: center;
gap: 10px;
flex-wrap: wrap;
}
.input-group label {
display: inline-block;
width: 150px;
font-weight: bold;
color: #555;
}
.input-group input[type="text"],
.input-group input[type="number"] {
flex-grow: 1;
padding: 10px 12px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 1rem;
box-sizing: border-box;
}
.input-group button {
padding: 10px 18px;
background-color: #004a99;
color: white;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 1rem;
transition: background-color 0.3s ease;
}
.input-group button:hover {
background-color: #003366;
}
#piecewiseFunctions {
width: 100%;
min-height: 150px;
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-family: 'Courier New', Courier, monospace;
font-size: 0.95rem;
box-sizing: border-box;
resize: vertical;
}
.output-section {
background-color: #eef5ff;
padding: 20px;
border-radius: 8px;
text-align: center;
border: 1px solid #cce0ff;
}
#result {
font-size: 1.5rem;
font-weight: bold;
color: #004a99;
margin-top: 10px;
word-wrap: break-word;
}
.article-section {
margin-top: 30px;
padding-top: 30px;
border-top: 1px solid #eee;
}
.article-section h3 {
text-align: left;
}
.article-section p, .article-section ul, .article-section li {
color: #555;
}
.article-section ul {
padding-left: 20px;
}
.button-container {
text-align: center;
margin-top: 20px;
}
#calculateBtn {
padding: 12px 25px;
background-color: #28a745;
color: white;
border: none;
border-radius: 5px;
cursor: pointer;
font-size: 1.1rem;
transition: background-color 0.3s ease;
}
#calculateBtn:hover {
background-color: #218838;
}
@media (max-width: 768px) {
.loan-calc-container {
flex-direction: column;
padding: 20px;
}
.input-group label {
width: auto;
margin-bottom: 5px;
}
.input-group input[type="text"],
.input-group input[type="number"] {
width: calc(100% – 10px);
}
}
Graph Data Output
Enter your function definitions and click "Generate Graph Data".
The output below contains points (x, y) that can be used to plot your function. You can copy this data and paste it into a graphing tool or a plotting library.
Understanding Piecewise Functions
A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval (or "piece") of the main function's domain. In simpler terms, it's like having several different rules for your function, and you use the appropriate rule depending on the input value.
Mathematical Definition
A piecewise function $f(x)$ can be formally defined as:
$$
f(x) =
\begin{cases}
g_1(x) & \text{if } x \in I_1 \\
g_2(x) & \text{if } x \in I_2 \\
\vdots & \vdots \\
g_n(x) & \text{if } x \in I_n
\end{cases}
$$
Here:
- $g_1(x), g_2(x), \dots, g_n(x)$ are the different function expressions.
- $I_1, I_2, \dots, I_n$ are the intervals of the domain over which each expression is valid. These intervals are typically defined using inequalities (e.g., $x < 0$, $x \ge 0$, $0 \le x < 5$).
How the Calculator Works
This calculator takes your definitions, which follow the format y = expression, condition, and interprets them. For each piece:
- Expression Parsing: It parses the mathematical expression (e.g.,
2*x + 1, x^2).
- Condition Evaluation: It parses the condition (e.g.,
x < 0, x >= 0).
- Point Generation: It generates a series of x-values within the specified X Min and X Max range.
- Conditional Application: For each generated x-value, it checks which condition it satisfies. If a condition is met, it evaluates the corresponding expression to find the y-value.
- Data Output: It outputs pairs of (x, y) coordinates that satisfy the function's definition.
The calculator aims to provide enough points to accurately represent the graph of each piece of the function across the specified domain.
Common Use Cases
- Mathematical Modeling: Representing real-world scenarios where different rules apply under different circumstances (e.g., tax brackets, utility pricing, speed limits that change based on time of day).
- Engineering: Designing systems with varying behaviors based on input parameters.
- Computer Graphics: Defining complex curves or shapes that are composed of simpler segments.
- Economics: Modeling cost functions, revenue, or profit under different production levels or market conditions.
- Education: Helping students visualize and understand the concept of piecewise functions in algebra and calculus.
Tips for Using the Calculator
- Ensure your conditions cover all parts of the domain you are interested in, or be aware that there might be gaps.
- Use standard mathematical notation (e.g.,
* for multiplication, ^ for exponentiation).
- Be precise with your inequality signs (
<, >, <=, >=).
- Check the generated data points to ensure they align with your expectations.
function calculatePiecewiseFunction() {
var functionDefinitionsInput = document.getElementById('piecewiseFunctions').value;
var xMin = parseFloat(document.getElementById('xMin').value);
var xMax = parseFloat(document.getElementById('xMax').value);
var numPoints = parseInt(document.getElementById('numPoints').value);
var resultDiv = document.getElementById('result');
if (isNaN(xMin) || isNaN(xMax) || isNaN(numPoints)) {
resultDiv.innerText = "Error: Please enter valid numbers for X Min, X Max, and Number of Points.";
return;
}
if (xMin >= xMax) {
resultDiv.innerText = "Error: X Min must be less than X Max.";
return;
}
if (numPoints < 10) {
resultDiv.innerText = "Error: Number of Points should be at least 10 for reasonable accuracy.";
return;
}
var lines = functionDefinitionsInput.split('\n');
var parsedFunctions = [];
for (var i = 0; i < lines.length; i++) {
var line = lines[i].trim();
if (!line) continue;
var parts = line.split(',');
if (parts.length !== 2) {
console.error("Skipping invalid line format: " + line);
continue;
}
var expressionPart = parts[0].trim();
var conditionPart = parts[1].trim();
// Basic validation for expression and condition
if (!expressionPart.toLowerCase().startsWith('y =') || !conditionPart.includes('x')) {
console.error("Skipping invalid format (expected 'y = expression, condition_with_x'): " + line);
continue;
}
var expression = expressionPart.substring(expressionPart.toLowerCase().indexOf('=') + 1).trim();
var condition = conditionPart;
try {
// Attempt to compile the function and condition evaluator
// Using Function constructor for dynamic evaluation (use with caution for untrusted input)
// We'll add 'Math.' prefix to common functions and constants for safety and availability
var compiledExpression = new Function('x', 'Math', `with(Math) { return ${expression}; }`);
var compiledCondition = new Function('x', `return ${condition};`);
parsedFunctions.push({ expression: compiledExpression, condition: compiledCondition, original: line });
} catch (e) {
console.error("Error parsing line: " + line, e);
resultDiv.innerText = `Error parsing line: "${line}". Check syntax.`;
return;
}
}
if (parsedFunctions.length === 0) {
resultDiv.innerText = "No valid function definitions found. Please check the format.";
return;
}
var allPoints = [];
var step = (xMax – xMin) / (numPoints – 1);
for (var j = 0; j < numPoints; j++) {
var x = xMin + j * step;
var y = NaN; // Default to NaN
var foundPiece = false;
for (var k = 0; k < parsedFunctions.length; k++) {
var funcData = parsedFunctions[k];
try {
if (funcData.condition(x)) {
y = funcData.expression(x, Math); // Pass Math object for common functions
// Ensure y is a finite number
if (isFinite(y)) {
allPoints.push({ x: x, y: y });
foundPiece = true;
break; // Stop after finding the first matching condition
} else {
// If expression results in Infinity or -Infinity, we might still want to record it as a point
// Or we can skip it if we only want finite values. For graphing, finite values are usually preferred.
// Let's skip non-finite results for simplicity in data output.
console.warn(`Non-finite result for x=${x} in function: ${funcData.original}`);
foundPiece = true; // Mark that a piece was found, even if result is not finite
break;
}
}
} catch (evalError) {
console.error(`Error evaluating condition or expression for x=${x} in function: ${funcData.original}`, evalError);
resultDiv.innerText = `Error evaluating function "${funcData.original}" at x=${x}. Check syntax and conditions.`;
return; // Stop calculation on error
}
}
// If no piece was found for this x, we don't add a point. This naturally creates gaps.
}
if (allPoints.length === 0) {
resultDiv.innerText = "No points generated. Ensure your conditions cover the range and are valid.";
} else {
// Format output as JSON array of objects
resultDiv.innerText = JSON.stringify(allPoints, null, 2);
}
}