Enter the function or data points to find the domain and range.
Function (Symbolic)
Set of Points (Ordered Pairs)
Understanding Domain and Range of a Graph
In mathematics, the domain and range are fundamental properties that describe a function or a relation represented by a graph. They tell us about the possible input values (domain) and output values (range) that the function can produce.
What is the Domain?
The domain of a function is the set of all possible input values (typically the 'x' values) for which the function is defined. In simpler terms, it's the set of all x-coordinates that you can find on the graph of the function.
Common restrictions for the domain include:
Division by zero: Functions with denominators cannot have inputs that make the denominator zero. For example, in f(x) = 1 / (x - 3), x cannot be 3.
Even roots: Functions involving even roots (like square roots) cannot have inputs that result in taking the root of a negative number. For example, in f(x) = sqrt(x - 2), x must be greater than or equal to 2 (x – 2 ≥ 0).
Logarithms: The argument of a logarithm must be strictly positive. For example, in f(x) = log(x), x must be greater than 0.
What is the Range?
The range of a function is the set of all possible output values (typically the 'y' values) that the function can produce. In simpler terms, it's the set of all y-coordinates that you can find on the graph of the function.
Determining the range often involves analyzing the behavior of the function. For example:
A horizontal line f(x) = c has a range of just {c}.
A parabola opening upwards like f(x) = x^2 has a range of [0, ∞).
A function like f(x) = sin(x) has a range of [-1, 1].
How to Use This Calculator
This calculator helps you find the domain and range in two ways:
Function Input: Enter a mathematical function (e.g., sqrt(x-2), 1/(x-5), log(x), sin(x), x^2). The calculator will attempt to determine the common symbolic domain and range based on standard mathematical rules.
Set of Points Input: Enter a series of ordered pairs (x,y). The calculator will determine the domain and range as the set of unique x-values and y-values from the points provided.
Note: For complex functions, symbolic computation of domain and range can be challenging. This calculator handles common cases. For advanced functions, graphical analysis or numerical methods might be required.
Examples
Example 1: Square Root Function
Input Function:sqrt(x - 2)
Expected Domain: [2, ∞) (because x – 2 must be ≥ 0)
Expected Range: [0, ∞) (the square root function outputs non-negative values)
Example 2: Rational Function
Input Function:1 / (x - 3)
Expected Domain: (-∞, 3) U (3, ∞) (because x cannot be 3)
Expected Range: (-∞, 0) U (0, ∞) (because y cannot be 0)
Example 3: Set of Points
Input Points:(1,5); (2,7); (3,5); (4,9)
Domain: {1, 2, 3, 4}
Range: {5, 7, 9}
function calculateDomainRange() {
var resultDiv = document.getElementById('result');
var errorMessageDiv = document.getElementById('errorMessage');
resultDiv.style.display = 'none';
errorMessageDiv.textContent = ";
var dataType = document.getElementById('dataType').value;
var domain = ";
var range = ";
if (dataType === 'function') {
var functionInput = document.getElementById('functionInput').value.trim();
if (!functionInput) {
errorMessageDiv.textContent = 'Please enter a function.';
return;
}
// — Symbolic Domain/Range Logic (Simplified for common cases) —
// This section requires more sophisticated parsing and mathematical reasoning
// For a practical implementation, consider a symbolic math library.
// Here, we implement basic checks for common function types.
if (functionInput.includes('sqrt(') || functionInput.includes('√(')) {
// Attempt to parse the argument of sqrt
var arg = ";
var openParen = functionInput.lastIndexOf('(');
var closeParen = functionInput.lastIndexOf(')');
if (openParen !== -1 && closeParen > openParen) {
arg = functionInput.substring(openParen + 1, closeParen);
}
if (arg.includes('x')) {
// Simple linear expression inside sqrt, e.g., x – a, x + a, ax + b
if (arg.includes('-')) {
var parts = arg.split('-');
if (parts.length === 2 && parts[1].trim() === '2') domain = '[' + parts[1].trim() + ', ∞)';
else if (parts.length === 2 && parts[0].trim() === 'x') domain = '[' + parts[1].trim() + ', ∞)';
else domain = 'Requires advanced parsing for ' + arg;
} else if (arg.includes('+')) {
var parts = arg.split('+');
if (parts.length === 2 && parts[0].trim() === 'x') domain = '[' + (parseFloat(parts[1].trim()) * -1) + ', ∞)';
else domain = 'Requires advanced parsing for ' + arg;
} else if (arg.trim() === 'x') {
domain = '[0, ∞)';
} else {
domain = 'Requires advanced parsing for ' + arg;
}
range = '[0, ∞)'; // Standard range for sqrt(positive_expression)
} else {
// If no 'x' in argument, e.g. sqrt(5), it's a constant.
var constantVal = parseFloat(arg);
if (!isNaN(constantVal) && constantVal >= 0) {
domain = 'All real numbers';
range = '[' + Math.sqrt(constantVal) + ', ∞)';
} else {
domain = 'No real solution';
range = 'No real solution';
}
}
} else if (functionInput.includes('1/')) {
// Rational function like 1/(x-a)
var denominator = ";
var slashIndex = functionInput.indexOf('1/');
if (slashIndex !== -1) {
denominator = functionInput.substring(slashIndex + 2).trim();
}
if (denominator.includes('x')) {
if (denominator.includes('-')) {
var parts = denominator.split('-');
if (parts.length === 2 && parts[0].trim() === 'x') {
var excludedValue = parseFloat(parts[1].trim());
if (!isNaN(excludedValue)) {
domain = '(-∞, ' + excludedValue + ') U (' + excludedValue + ', ∞)';
range = '(-∞, 0) U (0, ∞)'; // Standard for 1/(x-a)
} else {
domain = 'Requires advanced parsing for ' + denominator;
}
} else {
domain = 'Requires advanced parsing for ' + denominator;
}
} else if (denominator.includes('+')) {
var parts = denominator.split('+');
if (parts.length === 2 && parts[0].trim() === 'x') {
var excludedValue = parseFloat(parts[1].trim()) * -1;
if (!isNaN(excludedValue)) {
domain = '(-∞, ' + excludedValue + ') U (' + excludedValue + ', ∞)';
range = '(-∞, 0) U (0, ∞)'; // Standard for 1/(x+a)
} else {
domain = 'Requires advanced parsing for ' + denominator;
}
} else {
domain = 'Requires advanced parsing for ' + denominator;
}
} else if (denominator.trim() === 'x') {
domain = '(-∞, 0) U (0, ∞)';
range = '(-∞, 0) U (0, ∞)';
} else {
domain = 'Requires advanced parsing for ' + denominator;
}
} else {
// Constant in denominator, e.g., 1/5
var constantVal = parseFloat(denominator);
if (constantVal !== 0) {
domain = 'All real numbers';
range = '[' + (1 / constantVal) + ']';
} else {
domain = 'No real solution (division by zero)';
range = 'No real solution';
}
}
} else if (functionInput.toLowerCase().includes('log(')) {
var arg = ";
var openParen = functionInput.lastIndexOf('(');
var closeParen = functionInput.lastIndexOf(')');
if (openParen !== -1 && closeParen > openParen) {
arg = functionInput.substring(openParen + 1, closeParen);
}
if (arg.trim() === 'x') {
domain = '(0, ∞)';
range = 'All real numbers';
} else {
domain = 'Requires advanced parsing for log argument: ' + arg;
range = 'Requires advanced parsing';
}
} else if (functionInput.toLowerCase().includes('sin(x)') || functionInput.toLowerCase().includes('cos(x)')) {
domain = 'All real numbers';
range = '[-1, 1]';
} else if (functionInput.includes('^2') || functionInput.includes('**2')) { // Basic quadratic x^2
if (functionInput.trim() === 'x^2' || functionInput.trim() === 'x**2') {
domain = 'All real numbers';
range = '[0, ∞)';
} else {
domain = 'Requires advanced parsing for quadratic';
range = 'Requires advanced parsing';
}
} else {
// Default for simple linear functions or if parsing fails
if (functionInput.includes('x')) {
domain = 'All real numbers';
range = 'All real numbers';
} else {
// It's a constant function
var constantVal = parseFloat(functionInput);
if (!isNaN(constantVal)) {
domain = 'All real numbers';
range = '[' + constantVal + ']';
} else {
domain = 'Could not determine';
range = 'Could not determine';
}
}
}
if (domain === " || range === ") {
errorMessageDiv.textContent = 'Could not automatically determine domain/range for this function. Try the points input or simplify the function.';
resultDiv.style.display = 'none';
} else {
resultDiv.innerHTML = 'Domain: ' + domain + 'Range: ' + range;
resultDiv.style.display = 'block';
}
} else if (dataType === 'points') {
var pointsInput = document.getElementById('pointsInput').value.trim();
if (!pointsInput) {
errorMessageDiv.textContent = 'Please enter the points in (x,y) format.';
return;
}
var pointsArray = pointsInput.split(';');
var xValues = [];
var yValues = [];
var validPoints = true;
for (var i = 0; i < pointsArray.length; i++) {
var pointStr = pointsArray[i].trim();
if (pointStr.length < 3) continue; // Skip empty or invalid segments
var match = pointStr.match(/^\s*\(\s*(-?\d*\.?\d+)\s*,\s*(-?\d*\.?\d+)\s*\)\s*$/);
if (match) {
var x = parseFloat(match[1]);
var y = parseFloat(match[2]);
if (!isNaN(x) && !isNaN(y)) {
xValues.push(x);
yValues.push(y);
} else {
validPoints = false;
break;
}
} else {
validPoints = false;
break;
}
}
if (!validPoints || xValues.length === 0) {
errorMessageDiv.textContent = 'Invalid point format. Please use (x,y) pairs separated by semicolons (e.g., (1,2); (3,4)).';
return;
}
// Get unique values and sort for domain/range
var uniqueX = […new Set(xValues)].sort(function(a, b) { return a – b; });
var uniqueY = […new Set(yValues)].sort(function(a, b) { return a – b; });
// Format for display
domain = '{' + uniqueX.join(', ') + '}';
range = '{' + uniqueY.join(', ') + '}';
resultDiv.innerHTML = 'Domain: ' + domain + 'Range: ' + range;
resultDiv.style.display = 'block';
}
}
// Toggle visibility for points input based on data type selection
document.getElementById('dataType').addEventListener('change', function() {
var pointsInputGroup = document.getElementById('pointsInputGroup');
if (this.value === 'points') {
pointsInputGroup.style.display = 'block';
} else {
pointsInputGroup.style.display = 'none';
}
});