Select a function type and enter its parameters to determine its domain and range.
Linear: f(x) = mx + b
Quadratic: f(x) = ax² + bx + c
Square Root: f(x) = √(ax + b)
Rational: f(x) = 1 / (ax + b)
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font-family: 'Arial', sans-serif;
background-color: #f9f9f9;
padding: 20px;
border-radius: 8px;
box-shadow: 0 2px 10px rgba(0, 0, 0, 0.1);
max-width: 600px;
margin: 20px auto;
border: 1px solid #ddd;
}
.calculator-container h2 {
color: #333;
text-align: center;
margin-bottom: 20px;
font-size: 1.8em;
}
.calculator-container p {
color: #555;
text-align: center;
margin-bottom: 25px;
line-height: 1.6;
}
.calc-input-group {
margin-bottom: 15px;
padding: 10px;
background-color: #fff;
border-radius: 5px;
border: 1px solid #eee;
}
.calc-input-group label {
display: block;
margin-bottom: 8px;
color: #333;
font-weight: bold;
font-size: 0.95em;
}
.calc-input-group input[type="number"],
.calc-input-group select {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 1em;
box-sizing: border-box;
margin-top: 5px;
}
.calc-input-group input[type="number"]:focus,
.calc-input-group select:focus {
border-color: #007bff;
outline: none;
box-shadow: 0 0 5px rgba(0, 123, 255, 0.2);
}
.calculate-button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #007bff;
color: white;
border: none;
border-radius: 4px;
font-size: 1.1em;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 20px;
}
.calculate-button:hover {
background-color: #0056b3;
}
.calculator-result {
margin-top: 25px;
padding: 20px;
background-color: #e9f7ef;
border: 1px solid #d4edda;
border-radius: 8px;
color: #155724;
font-size: 1.1em;
line-height: 1.8;
}
.calculator-result h3 {
color: #0f5132;
margin-top: 0;
margin-bottom: 15px;
font-size: 1.4em;
}
.calculator-result p {
margin-bottom: 10px;
text-align: left;
}
.calculator-result strong {
color: #0f5132;
}
function showHideInputs() {
var functionType = document.getElementById("functionType").value;
document.getElementById("linearInputs").style.display = "none";
document.getElementById("quadraticInputs").style.display = "none";
document.getElementById("sqrtInputs").style.display = "none";
document.getElementById("rationalInputs").style.display = "none";
if (functionType === "linear") {
document.getElementById("linearInputs").style.display = "block";
} else if (functionType === "quadratic") {
document.getElementById("quadraticInputs").style.display = "block";
} else if (functionType === "sqrt") {
document.getElementById("sqrtInputs").style.display = "block";
} else if (functionType === "rational") {
document.getElementById("rationalInputs").style.display = "block";
}
}
function calculateDomainRange() {
var functionType = document.getElementById("functionType").value;
var domain = "";
var range = "";
var functionDisplay = "";
var explanation = "";
var resultDiv = document.getElementById("result");
resultDiv.innerHTML = ""; // Clear previous results
if (functionType === "linear") {
var m = parseFloat(document.getElementById("mValue").value);
var b = parseFloat(document.getElementById("bValueLinear").value);
if (isNaN(m) || isNaN(b)) {
resultDiv.innerHTML = "Please enter valid numbers for m and b.";
return;
}
functionDisplay = "f(x) = " + (m === 1 ? "" : (m === -1 ? "-" : m)) + "x " + (b >= 0 ? "+" : "") + b;
if (m === 0) {
functionDisplay = "f(x) = " + b;
domain = "(-∞, ∞)";
range = "[" + b + ", " + b + "]";
explanation = "For a constant function f(x) = b, the domain is all real numbers, and the range is just the constant value b.";
} else {
domain = "(-∞, ∞)";
range = "(-∞, ∞)";
explanation = "For a linear function f(x) = mx + b where m ≠ 0, there are no restrictions on x, so the domain is all real numbers. The function can output any real number, so the range is also all real numbers.";
}
} else if (functionType === "quadratic") {
var a = parseFloat(document.getElementById("aValueQuad").value);
var b = parseFloat(document.getElementById("bValueQuad").value);
var c = parseFloat(document.getElementById("cValueQuad").value);
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultDiv.innerHTML = "Please enter valid numbers for a, b, and c.";
return;
}
if (a === 0) {
// If a is 0, it's a linear function
functionDisplay = "f(x) = " + (b === 1 ? "" : (b === -1 ? "-" : b)) + "x " + (c >= 0 ? "+" : "") + c;
if (b === 0) {
functionDisplay = "f(x) = " + c;
domain = "(-∞, ∞)";
range = "[" + c + ", " + c + "]";
explanation = "When a=0 and b=0, the function becomes a constant function f(x) = c. The domain is all real numbers, and the range is just the constant value c.";
} else {
domain = "(-∞, ∞)";
range = "(-∞, ∞)";
explanation = "When a=0, the function becomes a linear function f(x) = bx + c. There are no restrictions on x, so the domain is all real numbers. The function can output any real number, so the range is also all real numbers.";
}
} else {
functionDisplay = "f(x) = " + (a === 1 ? "" : (a === -1 ? "-" : a)) + "x² " + (b >= 0 ? "+" : "") + b + "x " + (c >= 0 ? "+" : "") + c;
domain = "(-∞, ∞)"; // All real numbers for quadratic functions
// Calculate vertex y-coordinate for range
var vertexY = c – (b * b) / (4 * a);
if (a > 0) {
range = "[" + vertexY.toFixed(4) + ", ∞)";
explanation = "For a quadratic function f(x) = ax² + bx + c, the domain is all real numbers. Since a > 0, the parabola opens upwards, and the minimum value is at the vertex's y-coordinate. The range starts from this minimum value up to infinity.";
} else { // a < 0
range = "(-∞, " + vertexY.toFixed(4) + "]";
explanation = "For a quadratic function f(x) = ax² + bx + c, the domain is all real numbers. Since a = 0 ? "+" : "") + b + ")";
if (a === 0) {
if (b 0) {
domain = "[" + criticalPoint.toFixed(4) + ", ∞)";
explanation = "For a square root function f(x) = √(ax + b), the expression inside the square root must be non-negative (ax + b ≥ 0). Since a > 0, this means x ≥ -b/a. The range of a principal square root function is always non-negative.";
} else { // a < 0
domain = "(-∞, " + criticalPoint.toFixed(4) + "]";
explanation = "For a square root function f(x) = √(ax + b), the expression inside the square root must be non-negative (ax + b ≥ 0). Since a = 0 ? "+" : "") + b + ")";
if (a === 0) {
if (b === 0) {
domain = "No real domain";
range = "No real range";
explanation = "If a=0 and b=0, the function becomes 1/0, which is undefined.";
} else {
domain = "(-∞, ∞)";
range = "[" + (1/b).toFixed(4) + ", " + (1/b).toFixed(4) + "]";
explanation = "If a=0, the function becomes a constant f(x) = 1/b. The domain is all real numbers, and the range is just the constant value 1/b.";
}
} else {
var excludedX = -b / a;
domain = "(-∞, " + excludedX.toFixed(4) + ") U (" + excludedX.toFixed(4) + ", ∞)";
range = "(-∞, 0) U (0, ∞)";
explanation = "For a rational function f(x) = 1 / (ax + b), the denominator cannot be zero (ax + b ≠ 0), so x cannot be -b/a. The range excludes 0 because a fraction with a non-zero numerator can never equal zero.";
}
}
resultDiv.innerHTML =
"
In mathematics, the domain of a function refers to the set of all possible input values (often 'x' values) for which the function is defined and produces a real output. The range of a function, on the other hand, is the set of all possible output values (often 'y' values or f(x)) that the function can produce.
Why are Domain and Range Important?
Understanding the domain and range is fundamental in algebra, calculus, and various scientific fields. It helps us:
Identify valid inputs: Knowing the domain prevents us from trying to evaluate a function at points where it's undefined (e.g., dividing by zero, taking the square root of a negative number).
Predict function behavior: The range tells us what kind of outputs to expect, which is crucial for graphing functions and solving equations.
Model real-world phenomena: In practical applications, the domain and range often represent physical constraints. For example, time cannot be negative, and distances cannot be imaginary.
Common Function Types and Their Domain/Range Rules:
1. Linear Functions: f(x) = mx + b
A linear function creates a straight line when graphed. Here, 'm' is the slope and 'b' is the y-intercept.
Domain: For any linear function where 'm' is not zero, there are no restrictions on the input 'x'. You can plug in any real number, and you'll get a real output. Thus, the domain is all real numbers, written as (-∞, ∞). If m=0, it becomes a constant function f(x) = b, and the domain is still all real numbers.
Range: If 'm' is not zero, the line extends infinitely in both positive and negative y-directions. So, the range is also all real numbers, (-∞, ∞). If m=0, the function is f(x) = b, meaning the output is always 'b'. In this case, the range is just the single value [b, b].
A quadratic function creates a parabola when graphed. 'a', 'b', and 'c' are constants, with 'a' not equal to zero.
Domain: Similar to linear functions, there are no restrictions on the input 'x' for quadratic functions. You can square any real number, multiply it, and add constants without issues. So, the domain is always all real numbers, (-∞, ∞).
Range: The range depends on whether the parabola opens upwards (a > 0) or downwards (a < 0), and the y-coordinate of its vertex.
If a > 0 (parabola opens upwards), the vertex is the minimum point. The range will be from the y-coordinate of the vertex up to positive infinity: [yvertex, ∞).
If a < 0 (parabola opens downwards), the vertex is the maximum point. The range will be from negative infinity up to the y-coordinate of the vertex: (-∞, yvertex].
The y-coordinate of the vertex can be found using the formula: yvertex = c – b² / (4a).
Example: For f(x) = x² – 4x + 3, a=1, b=-4, c=3. Vertex x = -(-4)/(2*1) = 2. Vertex y = (2)² – 4(2) + 3 = 4 – 8 + 3 = -1. Since a > 0, Domain: (-∞, ∞), Range: [-1, ∞).
3. Square Root Functions: f(x) = √(ax + b)
These functions involve a square root.
Domain: The key restriction for square root functions in real numbers is that the expression inside the square root (the radicand) cannot be negative. Therefore, we must have ax + b ≥ 0. Solving this inequality for 'x' will give you the domain.
If a > 0, then x ≥ -b/a. Domain: [-b/a, ∞).
If a < 0, then x ≤ -b/a. Domain: (-∞, -b/a].
Range: The principal (positive) square root of a non-negative number is always non-negative. Thus, the range for f(x) = √(ax + b) is always [0, ∞).
Example: For f(x) = √(x – 2), we need x – 2 ≥ 0, so x ≥ 2. Domain: [2, ∞). Range: [0, ∞).
4. Rational Functions: f(x) = 1 / (ax + b)
Rational functions involve a ratio of two polynomials, where the denominator cannot be zero.
Domain: The primary restriction for rational functions is that the denominator cannot be equal to zero. So, we set the denominator to zero and exclude that value from the domain: ax + b ≠ 0. Solving for 'x' gives x ≠ -b/a. The domain is all real numbers except for this excluded value, written as (-∞, -b/a) U (-b/a, ∞).
Range: For a simple rational function like f(x) = 1 / (ax + b), the numerator is a non-zero constant. This means the function's output can never be zero. Therefore, the range is all real numbers except zero: (-∞, 0) U (0, ∞).
Example: For f(x) = 1 / (x + 3), we need x + 3 ≠ 0, so x ≠ -3. Domain: (-∞, -3) U (-3, ∞). Range: (-∞, 0) U (0, ∞).
This calculator provides a quick way to apply these rules for common function types, helping you understand how parameters affect the domain and range.