Enter the functions and their corresponding intervals (domains). The calculator supports up to three pieces.
Your graph points will appear here.
Understanding and Graphing Piecewise Functions
A piecewise function is a function that is defined by multiple sub-functions, each applying to a certain interval of the main function's domain. Essentially, it's like having several different functions stitched together, but each piece only exists over a specific range of x-values.
Definition
A piecewise function can be represented as:
f(x) = {
g₁(x), if x ∈ [a₁, b₁)
g₂(x), if x ∈ [a₂, b₂)
...
g(x), if x ∈ [a, b)
}
Where:
f(x) is the overall piecewise function.
g(x) are the individual sub-functions (e.g., linear, quadratic, constant).
[a, b) represent the intervals (domains) over which each sub-function is active. a is the lower bound (inclusive, denoted by '[') and b is the upper bound (exclusive, denoted by ')').
The variable 'x' represents the input to the function.
How to Graph a Piecewise Function
Identify the Pieces and Intervals: Break down the function definition into each sub-function and its corresponding domain interval.
Graph Each Sub-function: For each sub-function g(x), graph it as if it were a normal function without any domain restrictions.
Apply Domain Restrictions: This is the crucial step. For each graphed sub-function, erase or ignore the parts that fall outside its designated interval [a, b).
Handle Endpoints:
If the interval starts with [a (inclusive), draw a closed/solid circle at the point (a, g(a)).
If the interval ends with b) (exclusive), draw an open/hollow circle at the point (b, g(b)).
Be mindful of the function's value exactly at the boundary. If the boundary point is included in the interval of another piece, the graph should connect or have a jump discontinuity.
Combine the Pieces: The resulting graph is the collection of all the valid segments from each sub-function.
Calculator Usage
This calculator helps visualize the points that make up your piecewise function. Enter each function expression (e.g., 2*x + 3, x^2, 5) and the start (inclusive) and end (exclusive) points of its domain. The calculator will then output a set of points that you can use to sketch the graph. It supports handling intervals like -Infinity and Infinity.
Common Use Cases
Mathematical Analysis: Understanding function behavior, continuity, and differentiability at boundary points.
Engineering: Modeling systems that change behavior based on certain thresholds (e.g., variable pricing, different operational modes).
Economics: Representing progressive tax systems or tiered pricing structures.
Physics: Describing motion or forces that change abruptly at specific times or positions.
Note: This calculator generates points to aid in graphing. For a visual representation, you would typically use a dedicated graphing tool or plot these points on graph paper.
function evaluateFunction(funcString, x) {
try {
// Replace common math functions and operators
funcString = funcString.replace(/sin/g, 'Math.sin');
funcString = funcString.replace(/cos/g, 'Math.cos');
funcString = funcString.replace(/tan/g, 'Math.tan');
funcString = funcString.replace(/sqrt/g, 'Math.sqrt');
funcString = funcString.replace(/log/g, 'Math.log');
funcString = funcString.replace(/abs/g, 'Math.abs');
funcString = funcString.replace(/\^/g, '**'); // Support for exponentiation operator
// Create a safe evaluation context
var scope = {
x: x,
Math: Math
};
// Get keys for the scope
var scopeKeys = Object.keys(scope);
var scopeValues = scopeKeys.map(function(key) {
return scope[key];
});
// Use Function constructor for safer evaluation
var func = new Function(scopeKeys, 'return ' + funcString);
return func.apply(null, scopeValues);
} catch (e) {
console.error("Error evaluating function '" + funcString + "' at x=" + x + ":", e);
return NaN; // Return NaN on error
}
}
function isNumeric(n) {
return !isNaN(parseFloat(n)) && isFinite(n);
}
function parseDomainValue(value) {
if (value.toLowerCase() === '-infinity') {
return -Infinity;
} else if (value.toLowerCase() === 'infinity') {
return Infinity;
} else if (isNumeric(value)) {
return parseFloat(value);
}
return NaN; // Invalid value
}
function calculatePiecewisePoints() {
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = "Calculating…";
resultDiv.className = ""; // Reset class
var functions = [
document.getElementById("func1").value,
document.getElementById("func2").value,
document.getElementById("func3").value
];
var domains = [
{ start: document.getElementById("domain1_start").value, end: document.getElementById("domain1_end").value },
{ start: document.getElementById("domain2_start").value, end: document.getElementById("domain2_end").value },
{ start: document.getElementById("domain3_start").value, end: document.getElementById("domain3_end").value }
];
var points = [];
var errors = [];
var xStep = 0.5; // Step for sampling points within intervals
var numPointsPerInterval = 50; // Number of points to generate within finite intervals
for (var i = 0; i = domainEnd) {
errors.push("Invalid domain for Function " + (i + 1) + ": Start value must be less than End value.");
continue;
}
// Generate points within the interval
if (isFinite(domainStart) && isFinite(domainEnd)) {
var step = (domainEnd – domainStart) / numPointsPerInterval;
if (step <= 0) step = xStep; // Ensure a valid step if interval is very small
for (var x = domainStart; x = domainEnd) break;
var y = evaluateFunction(funcString, x);
if (!isNaN(y)) {
points.push({ x: x, y: y, funcIndex: i });
}
}
// Add the end point if it's finite and valid, and not already covered
if (isFinite(domainEnd)) {
var y_end = evaluateFunction(funcString, domainEnd);
if (!isNaN(y_end)) {
// Check if a point extremely close to domainEnd already exists
var exists = points.some(p => Math.abs(p.x – domainEnd) 0 ? -100 : -1000); // Adjust sampling range as needed
var sampleEnd = isFinite(domainEnd) ? domainEnd : (domainStart 0 ? -100 : -1000;
if (!isFinite(domainEnd)) sampleEnd = domainStart domainStart) sampleStart = domainStart;
if (isFinite(domainEnd) && sampleEnd < domainEnd) sampleEnd = domainEnd;
var step = (sampleEnd – sampleStart) / numPointsPerInterval;
if (step <= 0) step = xStep;
for (var x = sampleStart; x < sampleEnd; x += step) {
var effectiveX = x;
// Clip or extend to domain boundaries if necessary
if (isFinite(domainStart) && effectiveX = domainEnd) continue;
var y = evaluateFunction(funcString, effectiveX);
if (!isNaN(y)) {
points.push({ x: effectiveX, y: y, funcIndex: i });
}
}
// Add boundary points if they are finite and within the overall potential range
if (isFinite(domainStart)) {
var y_start = evaluateFunction(funcString, domainStart);
if (!isNaN(y_start)) {
var exists = points.some(p => Math.abs(p.x – domainStart) Math.abs(p.x – domainEnd) 0) {
resultDiv.innerHTML = errors.join("");
resultDiv.className = "error";
} else if (points.length === 0) {
resultDiv.innerHTML = "No points generated. Check your function expressions and domain intervals.";
resultDiv.className = "error";
} else {
// Sort points primarily by x-value for better visualization order
points.sort(function(a, b) {
return a.x – b.x;
});
var outputHTML = "
Generated Graph Points (x, y):
";
points.forEach(function(p) {
var label = "( " + p.x.toFixed(3) + ", " + p.y.toFixed(3) + " )";
// Optionally add function index or color coding if needed for complex graphs
outputHTML += "
" + label + "
";
});
outputHTML += "
";
outputHTML += "Note: Points generated by sampling. Use these to sketch your graph.";
resultDiv.innerHTML = outputHTML;
}
}