function calculateCombinedFunction() {
var functionFStr = document.getElementById('functionF').value;
var functionGStr = document.getElementById('functionG').value;
var xValueStr = document.getElementById('xValue').value;
var resultDiv = document.getElementById('result');
resultDiv.innerHTML = "; // Clear previous results
if (!functionFStr || !functionGStr) {
resultDiv.innerHTML = 'Please enter both function expressions.';
return;
}
var x = parseFloat(xValueStr);
if (isNaN(x)) {
resultDiv.innerHTML = 'Please enter a valid number for x.';
return;
}
// Replace common math notations for JavaScript compatibility
// e.g., x^2 -> Math.pow(x, 2) or x**2
// For simplicity and common use, we'll allow `**` for power.
// Users should be aware of JS math syntax (e.g., `Math.sqrt(x)`, `Math.sin(x)`)
var f_func, g_func;
try {
// Using new Function for dynamic function creation.
// This is powerful but requires careful input sanitization in a production environment.
// For a client-side calculator where user inputs are not stored/sent to server, it's common.
f_func = new Function('x', 'return ' + functionFStr.replace(/\^/g, '**'));
g_func = new Function('x', 'return ' + functionGStr.replace(/\^/g, '**'));
} catch (e) {
resultDiv.innerHTML = 'Error parsing function expressions. Please check syntax. (e.g., use `*` for multiplication, `**` for power, `Math.sqrt(x)` for square root).';
return;
}
var f_val, g_val;
try {
f_val = f_func(x);
g_val = g_func(x);
} catch (e) {
resultDiv.innerHTML = 'Error evaluating functions at x=' + x + '. This might be due to an invalid operation (e.g., division by zero within the function, sqrt of negative number).';
return;
}
if (isNaN(f_val) || isNaN(g_val)) {
resultDiv.innerHTML = 'One or both functions resulted in an undefined value (NaN) at x=' + x + '. Check for operations like square root of negative numbers or logarithms of non-positive numbers.';
return;
}
var operation = document.querySelector('input[name="operation"]:checked').value;
var finalResult;
var operationSymbol = ";
switch (operation) {
case 'f_plus_g':
finalResult = f_val + g_val;
operationSymbol = '(f + g)(' + x + ')';
break;
case 'f_minus_g':
finalResult = f_val – g_val;
operationSymbol = '(f – g)(' + x + ')';
break;
case 'f_times_g':
finalResult = f_val * g_val;
operationSymbol = '(f * g)(' + x + ')';
break;
case 'f_div_g':
if (g_val === 0) {
resultDiv.innerHTML = 'Division by zero detected for (f / g)(x) when g(x) = 0.';
return;
}
finalResult = f_val / g_val;
operationSymbol = '(f / g)(' + x + ')';
break;
case 'f_of_g':
try {
finalResult = f_func(g_val);
} catch (e) {
resultDiv.innerHTML = 'Error evaluating f(g(x)). This might be due to an invalid operation within f(x) when evaluated at g(x) = ' + g_val + '.';
return;
}
operationSymbol = 'f(g(' + x + '))';
break;
case 'g_of_f':
try {
finalResult = g_func(f_val);
} catch (e) {
resultDiv.innerHTML = 'Error evaluating g(f(x)). This might be due to an invalid operation within g(x) when evaluated at f(x) = ' + f_val + '.';
return;
}
operationSymbol = 'g(f(' + x + '))';
break;
default:
resultDiv.innerHTML = 'Unknown operation selected.';
return;
}
if (isNaN(finalResult)) {
resultDiv.innerHTML = 'The combined function resulted in an undefined value (NaN). Check the domain of the combined function.';
return;
}
resultDiv.innerHTML =
'For x =
';
}
.calculator-container {
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 {
text-align: center;
color: #333;
margin-bottom: 20px;
font-size: 1.8em;
}
.calc-input-group {
margin-bottom: 15px;
}
.calc-input-group label {
display: block;
margin-bottom: 5px;
color: #555;
font-weight: bold;
}
.calc-input-group input[type="text"],
.calc-input-group input[type="number"] {
width: calc(100% – 22px);
padding: 10px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 1em;
}
.radio-group {
display: flex;
flex-wrap: wrap;
gap: 10px;
margin-top: 5px;
}
.radio-group input[type="radio"] {
margin-right: 5px;
}
.radio-group label {
font-weight: normal;
color: #333;
margin-bottom: 0;
display: inline-flex;
align-items: center;
}
.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;
}
.calc-result {
background-color: #e9ecef;
padding: 15px;
border-radius: 4px;
margin-top: 20px;
border: 1px solid #dee2e6;
min-height: 50px;
}
.calc-result p {
margin: 5px 0;
color: #333;
font-size: 1.1em;
}
.calc-result p strong {
color: #000;
}
.calc-result .error {
color: #dc3545;
font-weight: bold;
}
Understanding Combined Functions
In mathematics, combining functions involves performing arithmetic operations or composition on two or more functions to create a new function. This process allows us to build more complex functions from simpler ones, and it's a fundamental concept in algebra and calculus.
Types of Combinations
Let's consider two functions, f(x) and g(x). We can combine them in several ways:
1. Sum of Functions: (f + g)(x) = f(x) + g(x)
The sum of two functions is obtained by adding their outputs for a given input x. The domain of (f + g)(x) is the intersection of the domains of f(x) and g(x).
Example: If f(x) = x^2 and g(x) = x + 3, then (f + g)(x) = x^2 + x + 3.
If x = 2:
f(2) = 2^2 = 4g(2) = 2 + 3 = 5(f + g)(2) = f(2) + g(2) = 4 + 5 = 9
2. Difference of Functions: (f – g)(x) = f(x) – g(x)
The difference of two functions is found by subtracting the output of g(x) from the output of f(x) for a given input x. The domain is also the intersection of the domains of f(x) and g(x).
Example: If f(x) = x^2 and g(x) = x + 3, then (f - g)(x) = x^2 - (x + 3) = x^2 - x - 3.
If x = 2:
f(2) = 4g(2) = 5(f - g)(2) = f(2) - g(2) = 4 - 5 = -1
3. Product of Functions: (f * g)(x) = f(x) * g(x)
The product of two functions is obtained by multiplying their outputs for a given input x. The domain is the intersection of the domains of f(x) and g(x).
Example: If f(x) = x^2 and g(x) = x + 3, then (f * g)(x) = x^2 * (x + 3) = x^3 + 3x^2.
If x = 2:
f(2) = 4g(2) = 5(f * g)(2) = f(2) * g(2) = 4 * 5 = 20
4. Quotient of Functions: (f / g)(x) = f(x) / g(x)
The quotient of two functions is found by dividing the output of f(x) by the output of g(x). The domain is the intersection of the domains of f(x) and g(x), with the additional restriction that g(x) cannot be zero.
Example: If f(x) = x^2 and g(x) = x + 3, then (f / g)(x) = x^2 / (x + 3).
If x = 2:
f(2) = 4g(2) = 5(f / g)(2) = f(2) / g(2) = 4 / 5 = 0.8
Note: If
x = -3,
g(-3) = 0, so
(f / g)(-3) would be undefined.
5. Composition of Functions: (f o g)(x) = f(g(x))
Function composition means applying one function to the result of another. For f(g(x)), you first evaluate g(x), and then use that result as the input for f(x). The domain of f(g(x)) consists of all x in the domain of g such that g(x) is in the domain of f.
Example: If f(x) = x^2 and g(x) = x + 3, then f(g(x)) = f(x + 3) = (x + 3)^2 = x^2 + 6x + 9.
If x = 2:
- First,
g(2) = 2 + 3 = 5
- Then,
f(g(2)) = f(5) = 5^2 = 25
6. Composition of Functions: (g o f)(x) = g(f(x))
Similarly, for g(f(x)), you first evaluate f(x), and then use that result as the input for g(x). The domain of g(f(x)) consists of all x in the domain of f such that f(x) is in the domain of g.
Example: If f(x) = x^2 and g(x) = x + 3, then g(f(x)) = g(x^2) = x^2 + 3.
If x = 2:
- First,
f(2) = 2^2 = 4
- Then,
g(f(2)) = g(4) = 4 + 3 = 7
Using the Calculator
Our calculator allows you to input two functions, f(x) and g(x), and a specific value for x. You can then select the desired operation (sum, difference, product, quotient, or composition) to see the result of the combined function evaluated at that x value.
Remember to use standard JavaScript mathematical syntax for your functions. For example:
- Multiplication:
* (e.g., 2*x)
- Exponents:
** (e.g., x**2 for x^2)
- Square root:
Math.sqrt(x)
- Trigonometric functions:
Math.sin(x), Math.cos(x), Math.tan(x)
- Logarithms:
Math.log(x) (natural log), Math.log10(x) (base 10)
Be mindful of the domains of your functions, especially for division and square roots, as operations outside the domain will result in errors or undefined values.