Piecewise Function Calculator & Grapher
Function Piece 1
Closed [a, b]
Closed-Open [a, b)
Open-Closed (a, b]
Open (a, b)
Enter function pieces and intervals to see results.
Function Graph
Y-Axis
X-Axis
Understanding Piecewise Functions
A piecewise function is a function defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Think of it as a function that behaves differently in different "pieces" of its domain.
Mathematical Definition
A piecewise function f(x) can be represented as:
f(x) =
g1(x), if x is in I1
g2(x), if x is in I2
…
gn(x), if x is in In
g2(x), if x is in I2
…
gn(x), if x is in In
Where:
gi(x)are the individual sub-functions (e.g., linear, quadratic, trigonometric).Iiare the intervals of the domain (e.g., x < 2, -1 ≤ x < 5, x ≥ 0).
Interval Notation
The intervals specify the range of x-values for which each sub-function is valid:
[a, b]: Includes all x such that a ≤ x ≤ b.(a, b): Includes all x such that a < x < b.[a, b): Includes all x such that a ≤ x < b.(a, b]: Includes all x such that a < x ≤ b.(-Infinity, a]or(-∞, a]: Includes all x such that x ≤ a.[a, Infinity)or[a, ∞): Includes all x such that x ≥ a.(-Infinity, Infinity)or(-∞, ∞): The entire real number line.
Graphing Piecewise Functions
To graph a piecewise function:
- Graph each sub-function on its own coordinate plane.
- For each sub-function, erase the parts that fall outside its specified interval.
- Pay close attention to the interval endpoints:
- A closed endpoint (included in the interval, denoted by
[or]) is represented by a solid dot (•). - An open endpoint (not included, denoted by
(or)) is represented by an open circle (o). - Ensure the pieces connect correctly at the boundaries if they overlap or meet.
Use Cases
Piecewise functions are used in various fields to model real-world scenarios where conditions change:
- Economics: Tax brackets, pricing tiers for services.
- Physics: Modeling motion with varying velocities or forces.
- Engineering: Designing systems with different operational modes.
- Computer Science: Algorithms that change behavior based on input size.
- Everyday Life: Utility billing (different rates per usage tier), postage costs based on weight.
This calculator helps visualize these functions, making it easier to understand their behavior across different domains.
Function Piece ${newPieceIndex + 1}
Closed [a, b]
Closed-Open [a, b)
Open-Closed (a, b]
Open (a, b)
`;
container.appendChild(newPieceDiv);
pieceCount++;
// Update the 'start' of the new piece to the 'end' of the previous one if it's not the first piece
if (newPieceIndex > 0) {
var prevEndInput = document.getElementById('end' + (newPieceIndex – 1));
var newStartInput = document.getElementById('start' + newPieceIndex);
if (prevEndInput && newStartInput) {
// Check if previous end was Infinity, if so, don't override new start with Infinity
if (prevEndInput.value !== 'Infinity') {
newStartInput.value = prevEndInput.value;
}
}
}
// Hide remove button for the first piece if it was previously visible
var firstPieceRemoveButton = document.querySelector('#piece0 .remove-piece-button');
if(firstPieceRemoveButton) {
firstPieceRemoveButton.style.display = 'none';
}
}
function removePiece(index) {
var pieceDiv = document.getElementById('piece' + index);
if (pieceDiv && pieceCount > 1) {
pieceDiv.remove();
pieceCount–;
// If the removed piece was the first one, ensure the next one's remove button is hidden
if (index === 0) {
var nextPieceRemoveButton = document.querySelector('#piece0 .remove-piece-button');
if (nextPieceRemoveButton) {
nextPieceRemoveButton.style.display = 'none';
}
}
// Re-adjust start value of subsequent pieces if needed
for (var i = index; i 0) {
// Only update if the current start is still Infinity or if it's not set
if (currentStartInput.value === 'Infinity' || currentStartInput.value === ") {
if (prevEndInput.value !== 'Infinity') {
currentStartInput.value = prevEndInput.value;
}
}
}
}
} else if (pieceCount === 1) {
alert("You must have at least one function piece.");
}
}
function calculateAndGraph() {
var functions = [];
var intervals = [];
var intervalTypes = [];
var validInputs = true;
for (var i = 0; i = endVal && startStr !== '-Infinity' && endStr !== 'Infinity') {
alert("Start of interval must be less than the end of interval.");
validInputs = false;
break;
}
// Validate interval type consistency
if ((startStr === '-Infinity' && intervalType[0] === '[') || (endStr === 'Infinity' && intervalType[1] === ')')) {
alert(`Interval type mismatch for piece ${i + 1}: Cannot use '[' with -Infinity or ')' with Infinity.`);
validInputs = false;
break;
}
var interval = { start: startVal, end: endVal, startChar: intervalType[0], endChar: intervalType[1] };
functions.push(funcString);
intervals.push(interval);
}
if (!validInputs) {
return;
}
var xMin = parseFloat(document.getElementById('xMin').value);
var xMax = parseFloat(document.getElementById('xMax').value);
var yMin = parseFloat(document.getElementById('yMin').value);
var yMax = parseFloat(document.getElementById('yMax').value);
var resolution = parseInt(document.getElementById('resolution').value);
if (isNaN(xMin) || isNaN(xMax) || isNaN(yMin) || isNaN(yMax) || isNaN(resolution) || resolution = xMax || yMin >= yMax) {
alert("Graph range minimums must be less than maximums.");
return;
}
var resultDiv = document.getElementById('result');
resultDiv.innerHTML = "Calculating and graphing…";
drawGraph(functions, intervals, xMin, xMax, yMin, yMax, resolution);
}
function drawGraph(functions, intervals, xMin, xMax, yMin, yMax, resolution) {
var canvas = document.getElementById('piecewiseCanvas');
var ctx = canvas.getContext('2d');
var width = canvas.width;
var height = canvas.height;
// Clear previous drawing
ctx.clearRect(0, 0, width, height);
// Function to convert graph coordinates to canvas coordinates
function getCanvasCoords(x, y) {
var canvasX = ((x – xMin) / (xMax – xMin)) * width;
var canvasY = height – ((y – yMin) / (yMax – yMin)) * height;
return { x: canvasX, y: canvasY };
}
// Draw Axes
ctx.beginPath();
ctx.strokeStyle = '#cccccc'; // Lighter gray for axes
ctx.lineWidth = 1;
// Y-Axis (at x=0 if within view)
if (xMin = 0) {
var zeroX = getCanvasCoords(0, yMin).x;
ctx.moveTo(zeroX, 0);
ctx.lineTo(zeroX, height);
} else {
// Draw at edge if 0 is not in range
var edgeX = xMin < 0 ? 0 : width;
ctx.moveTo(edgeX, 0);
ctx.lineTo(edgeX, height);
}
// X-Axis (at y=0 if within view)
if (yMin = 0) {
var zeroY = getCanvasCoords(xMin, 0).y;
ctx.moveTo(0, zeroY);
ctx.lineTo(width, zeroY);
} else {
// Draw at edge if 0 is not in range
var edgeY = yMin < 0 ? height : 0;
ctx.moveTo(0, edgeY);
ctx.lineTo(width, edgeY);
}
ctx.stroke();
// Draw Grid Lines (optional, can be performance intensive)
// Add grid lines for major ticks if desired
// Plot Functions
var colors = ['#007bff', '#28a745', '#ffc107', '#dc3545', '#6f42c1', '#17a2b8', '#fd7e14'];
var pointsPerInterval = Math.max(2, Math.floor(((xMax – xMin) / resolution) * 10)); // Ensure at least 2 points
for (var i = 0; i = sampleEnd && interval.start !== interval.end) {
continue;
}
// Calculate step for sampling within the relevant range
var step = (sampleEnd – sampleStart) / pointsPerInterval;
for (var p = 0; p = yMin && y = yMin && y = yMin && y = xMin && startX = yMin && startY = xMin && endX = yMin && endY <= yMax) {
var endCoords = getCanvasCoords(endX, endY);
ctx.fillStyle = (interval.endChar === ']') ? colors[i % colors.length] : 'white';
ctx.strokeStyle = colors[i % colors.length];
ctx.lineWidth = 1;
ctx.beginPath();
ctx.arc(endCoords.x, endCoords.y, 4, 0, 2 * Math.PI);
ctx.fill();
ctx.stroke();
}
}
}
// Update result text
document.getElementById('result').innerHTML = "Graph generated successfully!";
}
// Initial setup to ensure the first piece's remove button is hidden
window.onload = function() {
var firstPieceRemoveButton = document.querySelector('#piece0 .remove-piece-button');
if(firstPieceRemoveButton) {
firstPieceRemoveButton.style.display = 'none';
}
};