Enter your functions f(x) and g(x), and an input value for x. The calculator will compute either f(g(x)) or g(f(x)).
Result:
function calculateComposition() {
var functionFStr = document.getElementById('functionF').value;
var functionGStr = document.getElementById('functionG').value;
var inputValueXStr = document.getElementById('inputValueX').value;
var resultDiv = document.getElementById('result');
resultDiv.innerHTML = "; // Clear previous results
if (!functionFStr || !functionGStr || !inputValueXStr) {
resultDiv.innerHTML = 'Please fill in all fields.';
return;
}
var x = parseFloat(inputValueXStr);
if (isNaN(x)) {
resultDiv.innerHTML = 'Input Value for x must be a valid number.';
return;
}
var f_func, g_func;
try {
// Create functions dynamically. Using new Function() is similar to eval() but in a function scope.
// WARNING: Using new Function() or eval() with untrusted input can be a security risk.
// For a controlled calculator where inputs are expected to be mathematical expressions, it's acceptable.
f_func = new Function('x', 'return ' + functionFStr + ';');
g_func = new Function('x', 'return ' + functionGStr + ';');
} catch (e) {
resultDiv.innerHTML = 'Error parsing functions. Please check your syntax. Example: "2*x + 1" not "2x+1″.';
return;
}
var compositionTypeFG = document.getElementById('compositionTypeFG').checked;
var finalResult;
var intermediateValue;
var compositionLabel;
try {
if (compositionTypeFG) {
// Calculate f(g(x))
compositionLabel = 'f(g(x))';
intermediateValue = g_func(x);
finalResult = f_func(intermediateValue);
resultDiv.innerHTML = 'Given x = ' + x + ':';
resultDiv.innerHTML += 'First, calculate g(x): g(' + x + ') = ' + functionGStr.replace(/x/g, '(' + x + ')') + ' = ' + intermediateValue + ";
resultDiv.innerHTML += 'Then, calculate f(g(x)): f(' + intermediateValue + ') = ' + functionFStr.replace(/x/g, '(' + intermediateValue + ')') + ' = ' + finalResult + '';
} else {
// Calculate g(f(x))
compositionLabel = 'g(f(x))';
intermediateValue = f_func(x);
finalResult = g_func(intermediateValue);
resultDiv.innerHTML = 'Given x = ' + x + ':';
resultDiv.innerHTML += 'First, calculate f(x): f(' + x + ') = ' + functionFStr.replace(/x/g, '(' + x + ')') + ' = ' + intermediateValue + ";
resultDiv.innerHTML += 'Then, calculate g(f(x)): g(' + intermediateValue + ') = ' + functionGStr.replace(/x/g, '(' + intermediateValue + ')') + ' = ' + finalResult + '';
}
} catch (e) {
resultDiv.innerHTML = 'Error during calculation. Please ensure your functions are valid for the given input. Details: ' + e.message + ";
return;
}
}
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font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f9f9f9;
border: 1px solid #ddd;
border-radius: 8px;
padding: 25px;
max-width: 600px;
margin: 30px auto;
box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08);
}
.calculator-container h2 {
text-align: center;
color: #333;
margin-bottom: 20px;
font-size: 26px;
}
.calculator-content p {
margin-bottom: 15px;
line-height: 1.6;
color: #555;
}
.form-group {
margin-bottom: 18px;
}
.form-group label {
display: block;
margin-bottom: 8px;
color: #333;
font-weight: bold;
}
.form-group input[type="text"],
.form-group input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 5px;
font-size: 16px;
box-sizing: border-box;
}
.form-group input[type="radio"] {
margin-right: 8px;
}
.form-group input[type="radio"] + label {
display: inline-block;
font-weight: normal;
margin-right: 15px;
}
button {
background-color: #007bff;
color: white;
padding: 12px 25px;
border: none;
border-radius: 5px;
font-size: 18px;
cursor: pointer;
display: block;
width: 100%;
box-sizing: border-box;
transition: background-color 0.3s ease;
}
button:hover {
background-color: #0056b3;
}
.result-container {
background-color: #e9ecef;
border: 1px solid #ced4da;
border-radius: 5px;
padding: 15px;
margin-top: 25px;
}
.result-container h3 {
color: #333;
margin-top: 0;
margin-bottom: 10px;
font-size: 20px;
}
.result-container p {
color: #333;
font-size: 16px;
margin-bottom: 8px;
}
.result-container strong {
color: #007bff;
font-size: 18px;
}
Understanding Function Composition
Function composition is a fundamental concept in mathematics where one function's output becomes the input of another function. It's like a chain reaction: you take an initial value, apply one function to it, and then take that result and apply a second function to it.
What is Function Composition?
Given two functions, f(x) and g(x), there are two primary ways to compose them:
f(g(x)) (read as "f of g of x"): In this composition, the function g(x) is applied first to the input 'x'. The output of g(x) then becomes the input for the function f(x).
g(f(x)) (read as "g of f of x"): Here, the function f(x) is applied first to the input 'x'. The output of f(x) then becomes the input for the function g(x).
It's crucial to understand that f(g(x)) is generally not the same as g(f(x)). The order of operations matters significantly.
How to Use the Calculator
Our Function Composition Calculator simplifies the process of evaluating composed functions for a specific input value. Follow these steps:
Enter Function f(x): Type the mathematical expression for your first function into the "Function f(x)" field. Use 'x' as your variable. For multiplication, explicitly use the '*' symbol (e.g., 2*x + 1, not 2x+1).
Enter Function g(x): Similarly, type the mathematical expression for your second function into the "Function g(x)" field.
Enter Input Value for x: Provide the numerical value for 'x' that you want to evaluate the composed function at.
Select Composition Type: Choose whether you want to calculate f(g(x)) or g(f(x)) using the radio buttons.
Click "Calculate Composition": The calculator will then display the step-by-step evaluation and the final result.
Examples of Function Composition
Let's illustrate with some common functions:
Example 1: Basic Polynomials
Let f(x) = 2x + 1
Let g(x) = x²
Let x = 3
Calculating f(g(x)):
First, find g(3): g(3) = 3² = 9
Then, find f(9): f(9) = 2*(9) + 1 = 18 + 1 = 19
So, f(g(3)) = 19.
Calculating g(f(x)):
First, find f(3): f(3) = 2*(3) + 1 = 6 + 1 = 7
Then, find g(7): g(7) = 7² = 49
So, g(f(3)) = 49.
As you can see, f(g(3)) ≠ g(f(3)).
Example 2: More Complex Functions
Let f(x) = sqrt(x + 5) (use Math.sqrt(x + 5) in the calculator)
Function composition is not just a theoretical concept; it has wide-ranging applications:
Computer Science: In programming, function composition is a powerful paradigm for building complex operations from simpler ones. It's central to functional programming.
Physics: Describing motion where one quantity depends on another, which in turn depends on time (e.g., position as a function of velocity, and velocity as a function of time).
Economics: Modeling scenarios where the cost of production depends on the number of units, and the number of units depends on demand.
Calculus: The Chain Rule for differentiation is directly based on the concept of function composition.
Real-world Modeling: Any situation where a process involves multiple sequential steps, and the output of one step feeds into the next, can be modeled using function composition.
This calculator provides a quick and easy way to explore and verify function compositions, helping you grasp this essential mathematical concept.