Enter the function in standard mathematical notation (e.g., 'sqrt(x-2)', '1/(x^2-4)', 'log(x+1)').
Supported functions: sqrt(), abs(), log(), ln(), sin(), cos(), tan(). Use standard operators +, -, *, /, ^ for powers.
Domain will be displayed here.
Understanding the Domain of a Function
In mathematics, the domain of a function is the set of all possible input values (often represented by the independent variable, like 'x') for which the function is defined and produces a real number output. In simpler terms, it's what you are "allowed" to put into the function.
Why is the Domain Important?
Identifying the domain is crucial for understanding a function's behavior, graphing it accurately, and solving mathematical problems. Certain mathematical operations have inherent restrictions that limit the possible inputs.
Common Restrictions Leading to Exclusions from the Domain:
Division by Zero: A function is undefined when its denominator is zero. For functions like f(x) = 1/g(x), we must ensure g(x) ≠ 0.
Even Roots of Negative Numbers: Functions involving even roots (like the square root, sqrt()) are undefined for negative inputs under the radical. For f(x) = sqrt(g(x)), we must ensure g(x) ≥ 0.
Logarithms of Non-Positive Numbers: The logarithm function (log() or ln()) is only defined for positive arguments. For f(x) = log(g(x)) or f(x) = ln(g(x)), we must ensure g(x) > 0.
How the Calculator Works:
This calculator attempts to identify common restrictions based on the structure of the function you input. It focuses on:
Detecting potential division by zero.
Identifying expressions under square roots (or other even roots).
Finding arguments of logarithmic functions.
It works by parsing the function string and applying rules based on the operations and functions detected. For complex or piecewise functions, manual analysis might still be necessary.
Example Use Cases:
f(x) = 1 / (x - 5): The denominator cannot be zero, so x - 5 ≠ 0. This means x ≠ 5. The domain is all real numbers except 5, often written as (-∞, 5) U (5, ∞).
f(x) = sqrt(x + 3): The expression under the square root must be non-negative, so x + 3 ≥ 0. This means x ≥ -3. The domain is [-3, ∞).
f(x) = log(x^2 - 9): The argument of the logarithm must be positive, so x^2 - 9 > 0. This implies (x - 3)(x + 3) > 0, which holds true when x < -3 or x > 3. The domain is (-∞, -3) U (3, ∞).
f(x) = 2x + 1: This is a linear function. There are no restrictions on the value of 'x'. The domain is all real numbers, often written as (-∞, ∞).
Limitations:
This calculator provides a basic analysis. It may not handle all intricate mathematical scenarios, such as:
Piecewise functions.
Functions involving inverse trigonometric functions (like arcsin, arccos) which have specific domain limitations.
Implicitly defined functions.
Higher-order roots (e.g., cube roots, which do not restrict negative inputs).
Always verify the results with your understanding of mathematical principles.
function calculateDomain() {
var functionString = document.getElementById("functionInput").value.trim();
var variable = document.getElementById("variableInput").value.trim() || 'x'; // Default to 'x' if empty
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = "Calculating…";
if (!functionString) {
resultDiv.innerHTML = "Please enter a function.";
return;
}
// Basic check for common issues
var domainRestrictions = [];
var originalFunctionString = functionString;
// Normalize function string for easier parsing (e.g., replace ^ with **)
functionString = functionString.replace(/\^/g, '**');
// — Rules for Restrictions —
// 1. Denominators (Division by zero)
var divisionRegex = /([^+\-*/\s(]+)\s*\/\s*([^+\-*/\s)]+)/g;
var match;
while ((match = divisionRegex.exec(originalFunctionString)) !== null) {
var denominator = match[2];
// Simplify the denominator if it's a simple variable or expression
if (denominator && denominator.indexOf(variable) !== -1) {
domainRestrictions.push({
type: 'division',
condition: denominator + " != 0″,
explanation: "Denominator cannot be zero: " + denominator + " ≠ 0″
});
}
}
// 2. Square Roots (Even roots)
var sqrtRegex = /sqrt\(\s*([^)]+)\s*\)/g;
while ((match = sqrtRegex.exec(originalFunctionString)) !== null) {
var radicand = match[1];
if (radicand && radicand.indexOf(variable) !== -1) {
domainRestrictions.push({
type: 'sqrt',
condition: radicand + " >= 0″,
explanation: "Expression under square root must be non-negative: " + radicand + " ≥ 0″
});
}
}
// 3. Logarithms (Natural and Base 10)
var logRegex = /(log|ln)\(\s*([^)]+)\s*\)/g;
while ((match = logRegex.exec(originalFunctionString)) !== null) {
var logArg = match[2];
if (logArg && logArg.indexOf(variable) !== -1) {
domainRestrictions.push({
type: 'log',
condition: logArg + " > 0″,
explanation: "Argument of logarithm must be positive: " + logArg + " > 0″
});
}
}
// — Process Restrictions —
if (domainRestrictions.length === 0) {
// Check if the function is just a constant or only involves constants/operations that don't restrict the domain
try {
// Attempt to evaluate the function at a dummy value, ignoring variable for now
// This is a simplified check. A robust check would require a full AST parser.
// If it doesn't throw an error for simple cases, assume it's all real numbers.
// Example: "5", "2*3", "sin(45)" (if degree mode was assumed, but we're not)
// This part is heuristic and might incorrectly label complex functions as R.
if (!/[+\-*/^()]|sqrt|log|ln|sin|cos|tan/.test(originalFunctionString) || /^[0-9.\s]+$/.test(originalFunctionString)) {
resultDiv.innerHTML = "Domain: All Real Numbers (-∞, ∞)";
} else {
// If there are operations but no explicit restrictions found, it's still likely R for basic functions.
// A more advanced parser would be needed for certainty.
resultDiv.innerHTML = "Domain: All Real Numbers (-∞, ∞) (Assuming no hidden restrictions)";
}
} catch (e) {
resultDiv.innerHTML = "Could not determine domain automatically. Manual analysis needed.";
}
} else {
var resultString = "Domain Restrictions:";
var conditionsMet = []; // Store conditions to display combined intervals
var uniqueExplanations = new Set();
domainRestrictions.forEach(function(restriction) {
if (!uniqueExplanations.has(restriction.explanation)) {
resultString += "- " + restriction.explanation + "";
uniqueExplanations.add(restriction.explanation);
conditionsMet.push(restriction.condition);
}
});
// Display a simplified interpretation. Actual interval notation requires solving inequalities.
if (conditionsMet.length > 0) {
resultString += "Analysis: The function is defined for all values of '" + variable + "' that satisfy all these conditions simultaneously. For complex functions, solving these inequalities might lead to interval notation (e.g., x ≥ 2, x ≠ 3 implies [2, 3) U (3, ∞)).";
resultDiv.innerHTML = resultString;
} else {
// This case might occur if restrictions were found but were redundant or simplified away.
resultDiv.innerHTML = "Domain: All Real Numbers (-∞, ∞) (Initial checks found no explicit restrictions)";
}
}
}