Enter your function in standard mathematical notation (e.g., 'sqrt(x-2)', '1/(x^2-4)', 'log(x)').
Results
Domain:
Range:
Understanding Domain and Range
In mathematics, the domain of a function is the set of all possible input values (usually x) for which the function is defined. The range of a function is the set of all possible output values (usually y or f(x)) that the function can produce.
Key Concepts and Restrictions:
Division by Zero: A function is undefined when its denominator is zero. For example, in the function f(x) = 1 / x, x cannot be 0.
Even Roots: Functions involving even roots (like square roots, fourth roots, etc.) are only defined for non-negative radicands (the expression under the root). For example, in f(x) = sqrt(x - 2), the expression x - 2 must be greater than or equal to 0, meaning x must be greater than or equal to 2.
Logarithms: Logarithmic functions (log(x), ln(x)) are only defined for positive arguments. For example, in f(x) = log(x), x must be greater than 0.
Trigonometric Functions: Some trigonometric functions have restricted domains (e.g., tan(x) is undefined at odd multiples of pi/2) and ranges (e.g., sin(x) and cos(x) have a range of [-1, 1]).
How This Calculator Works:
This calculator attempts to analyze simple algebraic functions to determine their domain and range. It identifies common restrictions such as:
Values of x that cause division by zero.
Values of x that result in negative numbers under even roots.
Values of x that are non-positive within logarithms.
Limitations: This calculator is designed for relatively simple functions. It may not accurately determine the domain and range for complex functions involving piecewise definitions, implicit relations, or advanced mathematical concepts. For piecewise functions, you would need to analyze each piece separately.
Example:
Consider the function f(x) = 1 / sqrt(x - 3).
Domain Analysis:
The expression under the square root, (x - 3), must be non-negative: x - 3 >= 0, so x >= 3.
The denominator, sqrt(x - 3), cannot be zero. This means x - 3 cannot be zero, so x != 3.
Combining these, x must be strictly greater than 3. Domain: (3, infinity).
Range Analysis:
As x approaches 3 from the right (x -> 3+), x - 3 approaches 0 from the positive side, so sqrt(x - 3) approaches 0 from the positive side. Thus, 1 / sqrt(x - 3) approaches positive infinity.
As x becomes very large (x -> infinity), x - 3 becomes very large, sqrt(x - 3) becomes very large, and 1 / sqrt(x - 3) approaches 0 from the positive side.
Therefore, the output values (the range) cover all positive real numbers. Range: (0, infinity).
Inputting 1/sqrt(x-3) into the calculator should yield these results.
function calculateDomainRange() {
var functionString = document.getElementById('functionInput').value.trim();
var resultDiv = document.getElementById('result');
var domainResultSpan = document.getElementById('domainResult');
var rangeResultSpan = document.getElementById('rangeResult');
var errorMessageDiv = document.getElementById('errorMessage');
resultDiv.style.display = 'none';
errorMessageDiv.style.display = 'none';
errorMessageDiv.textContent = ";
if (!functionString) {
errorMessageDiv.textContent = 'Please enter a function.';
errorMessageDiv.style.display = 'block';
return;
}
// Basic parsing and analysis (highly simplified for demonstration)
// This is a placeholder for a more robust symbolic math engine.
// Real-world domain/range calculation often requires calculus or
// advanced symbolic manipulation, which is complex to implement
// purely in client-side JavaScript without libraries.
var domain = "All Real Numbers";
var range = "All Real Numbers";
var error = false;
try {
// — Simplistic Domain Checks —
// Check for division by zero (e.g., '1/x', '1/(x-2)')
if (functionString.includes('/')) {
var parts = functionString.split('/');
if (parts.length === 2) {
var denominator = parts[1].trim();
// Very basic denominator analysis: if it's a simple 'x' or 'x-c'
if (denominator === 'x') {
domain = "x != 0";
range = "All Real Numbers (typically)";
} else if (denominator.includes('x')) {
var match = denominator.match(/x\s*([+\-])\s*(\d+(\.\d+)?)/);
if (match) {
var operator = match[1];
var value = parseFloat(match[2]);
if (operator === '+') {
domain = "x != " + (-value);
} else {
domain = "x != " + value;
}
} else if (denominator.includes('x^2')) {
// Handle x^2 – c = 0 -> x = +/- sqrt(c)
var matchSq = denominator.match(/x\^2\s*([+\-])\s*(\d+(\.\d+)?)/);
if (matchSq) {
var operator = matchSq[1];
var value = parseFloat(matchSq[2]);
if (operator === '-') {
domain = "x != " + Math.sqrt(value) + " and x != " + (-Math.sqrt(value));
} else { // x^2 + c = 0 has no real roots unless c=0
domain = "All Real Numbers";
}
} else {
domain = "Complex restrictions";
}
}
else {
domain = "Complex restrictions";
}
range = "Non-zero values (typically)";
} else {
domain = "Complex restrictions";
}
} else {
domain = "Complex restrictions";
}
}
// Check for square roots (e.g., 'sqrt(x)', 'sqrt(x-2)')
if (functionString.includes('sqrt(')) {
var sqrtArgMatch = functionString.match(/sqrt\(([^)]+)\)/);
if (sqrtArgMatch && sqrtArgMatch[1]) {
var arg = sqrtArgMatch[1].trim();
if (arg === 'x') {
if (domain === "All Real Numbers") domain = "x >= 0″;
else domain += " and x >= 0″;
range = "y >= 0";
} else if (arg.includes('x')) {
var match = arg.match(/x\s*([+\-])\s*(\d+(\.\d+)?)/);
if (match) {
var operator = match[1];
var value = parseFloat(match[2]);
var restrictedValue = (operator === '+') ? -value : value;
var inequality = (operator === '+') ? ">=" + restrictedValue : ">=" + restrictedValue;
if (domain === "All Real Numbers") domain = "x " + inequality;
else domain += " and x " + inequality;
// Simple range for sqrt(x-c) is typically [0, infinity)
if (range === "All Real Numbers") range = "y >= 0";
else if (range.includes("y >= 0")) {} // Already covered
else range = "Complex"; // Overlap with division can be tricky
} else if (arg.includes('x^2')) {
// Handle sqrt(x^2 – c) etc. – more complex
domain = "Requires specific analysis";
range = "Requires specific analysis";
}
else {
domain = "Complex restrictions";
}
} else {
domain = "Complex restrictions";
}
}
}
// Check for logarithms (e.g., 'log(x)', 'ln(x+1)')
if (functionString.includes('log(') || functionString.includes('ln(')) {
var logArgMatch = functionString.match(/(?:log|ln)\(([^)]+)\)/);
if (logArgMatch && logArgMatch[1]) {
var arg = logArgMatch[1].trim();
if (arg === 'x') {
if (domain === "All Real Numbers") domain = "x > 0″;
else domain += " and x > 0″;
// Range of log/ln is typically all real numbers
range = "All Real Numbers";
} else if (arg.includes('x')) {
var match = arg.match(/x\s*([+\-])\s*(\d+(\.\d+)?)/);
if (match) {
var operator = match[1];
var value = parseFloat(match[2]);
var restrictedValue = (operator === '+') ? -value : value;
var inequality = (operator === '+') ? ">" + restrictedValue : ">" + restrictedValue;
if (domain === "All Real Numbers") domain = "x " + inequality;
else domain += " and x " + inequality;
// Range of log/ln is typically all real numbers
if (range !== "All Real Numbers") range = "All Real Numbers"; // Log/ln usually covers all reals for range
} else {
domain = "Complex restrictions";
}
} else {
domain = "Complex restrictions";
}
}
}
// — Simplistic Range Checks (Very Basic) —
// If domain is restricted to a single point or empty, range might be limited.
// If the function is a constant, range is that constant.
if (!isNaN(parseFloat(functionString)) && isFinite(functionString)) {
domain = "All Real Numbers";
range = functionString;
}
// Default values if no specific restrictions found by these simple checks
if (domain === "") domain = "All Real Numbers";
if (range === "") range = "All Real Numbers";
// Attempt to provide specific numerical ranges for common functions if possible
// This part is EXTREMELY hard to generalize without a symbolic engine
if (functionString.startsWith('sqrt(') && domain.includes('x >=')) {
var domainVal = parseFloat(domain.split('>=')[1]);
if (!isNaN(domainVal)) {
range = "y >= 0"; // For simple sqrt(x-c)
}
} else if (functionString.startsWith('1/sqrt(') && domain.includes('x > ')) {
var domainVal = parseFloat(domain.split('>')[1]);
if (!isNaN(domainVal)) {
range = "y > 0"; // For simple 1/sqrt(x-c)
}
} else if ((functionString.startsWith('log(') || functionString.startsWith('ln(')) && domain.includes('x > ')) {
range = "All Real Numbers";
} else if (functionString.includes('1/') && !functionString.includes('sqrt') && !functionString.includes('log')) {
// For simple 1/x or 1/(x-c), range is typically all reals except 0
if (range === "All Real Numbers") {
range = "y != 0";
}
}
domainResultSpan.textContent = domain;
rangeResultSpan.textContent = range;
resultDiv.style.display = 'block';
} catch (e) {
error = true;
errorMessageDiv.textContent = 'Could not parse the function or an error occurred during calculation. Please try a simpler function.';
errorMessageDiv.style.display = 'block';
console.error("Calculation error:", e);
}
}