Enter functions and an x-value, then click 'Calculate Composition'.
function calculateComposition() {
var f_expr = document.getElementById('functionF').value;
var g_expr = document.getElementById('functionG').value;
var x_val_str = document.getElementById('xValue').value;
var compositionType = document.querySelector('input[name="compositionType"]:checked').value;
var resultDiv = document.getElementById('compositionResult');
if (!f_expr || !g_expr) {
resultDiv.innerHTML = "Please enter both function expressions (f(x) and g(x)).";
return;
}
var x_val = parseFloat(x_val_str);
var isXValueProvided = !isNaN(x_val_str) && x_val_str !== ";
if (!isXValueProvided) {
resultDiv.innerHTML = "Please provide a numerical value for 'x' to evaluate the composition. This calculator performs numerical evaluation only.";
return;
}
var finalResult = ";
try {
// Helper function to prepare and evaluate an expression
function evaluateExpression(expression, valueForX) {
var tempExpression = expression;
// Replace 'x' with the actual value, ensuring proper parenthesization for negative numbers
tempExpression = tempExpression.replace(/x/g, '(' + valueForX + ')');
// Replace common math functions and constants with Math. prefix
tempExpression = tempExpression.replace(/sin\(/g, 'Math.sin(');
tempExpression = tempExpression.replace(/cos\(/g, 'Math.cos(');
tempExpression = tempExpression.replace(/tan\(/g, 'Math.tan(');
tempExpression = tempExpression.replace(/sqrt\(/g, 'Math.sqrt(');
tempExpression = tempExpression.replace(/log\(/g, 'Math.log('); // Natural log
tempExpression = tempExpression.replace(/abs\(/g, 'Math.abs(');
tempExpression = tempExpression.replace(/pow\(/g, 'Math.pow(');
tempExpression = tempExpression.replace(/PI/g, 'Math.PI');
tempExpression = tempExpression.replace(/E/g, 'Math.E');
// Handle power operator `^` by replacing it with `**`
tempExpression = tempExpression.replace(/\^/g, '**');
// Evaluate the expression using a Function constructor for isolated scope
var func = new Function('return ' + tempExpression);
return func();
}
var innerResult;
var outerResult;
var compositionLabel = ";
if (compositionType === 'fgx') {
compositionLabel = 'f(g(x))';
innerResult = evaluateExpression(g_expr, x_val);
if (isNaN(innerResult) || !isFinite(innerResult)) {
resultDiv.innerHTML = "Error: Evaluation of g(" + x_val + ") resulted in an invalid or non-finite number. Check g(x) or x value.";
return;
}
outerResult = evaluateExpression(f_expr, innerResult);
if (isNaN(outerResult) || !isFinite(outerResult)) {
resultDiv.innerHTML = "Error: Evaluation of f(g(" + x_val + ")) resulted in an invalid or non-finite number. Check f(x) or the result of g(x).";
return;
}
finalResult = "f(g(" + x_val + ")) = f(" + innerResult + ") = " + outerResult;
} else { // gfx
compositionLabel = 'g(f(x))';
innerResult = evaluateExpression(f_expr, x_val);
if (isNaN(innerResult) || !isFinite(innerResult)) {
resultDiv.innerHTML = "Error: Evaluation of f(" + x_val + ") resulted in an invalid or non-finite number. Check f(x) or x value.";
return;
}
outerResult = evaluateExpression(g_expr, innerResult);
if (isNaN(outerResult) || !isFinite(outerResult)) {
resultDiv.innerHTML = "Error: Evaluation of g(f(" + x_val + ")) resulted in an invalid or non-finite number. Check g(x) or the result of f(x).";
return;
}
finalResult = "g(f(" + x_val + ")) = g(" + innerResult + ") = " + outerResult;
}
resultDiv.innerHTML = "
Result for " + compositionLabel + ":
" + finalResult + "";
} catch (e) {
resultDiv.innerHTML = "An error occurred during calculation. Please check your function expressions for syntax errors.Important:
Use `*` for multiplication (e.g., `2*x`).
Use `**` for powers (e.g., `x**2`) or `Math.pow(base, exponent)`.
Prefix mathematical functions with `Math.` (e.g., `Math.sin(x)`, `Math.sqrt(x)`, `Math.log(x)` for natural log).
Constants like `PI` and `E` should be `Math.PI` and `Math.E`.
Function composition is a fundamental concept in mathematics where one function is applied to the result of another function. It's like a two-step process: the output of the first function becomes the input for the second function.
Notation and Definition
The composition of two functions, say f and g, is denoted as (f o g)(x) or f(g(x)). This means you first evaluate the inner function g(x), and then you take that result and plug it into the outer function f(x).
(f o g)(x) = f(g(x)): Evaluate g(x) first, then apply f to the result.
(g o f)(x) = g(f(x)): Evaluate f(x) first, then apply g to the result.
It's crucial to understand that the order of composition matters; f(g(x)) is generally not the same as g(f(x)).
Domain of Composite Functions
The domain of a composite function f(g(x)) consists of all values of x in the domain of g such that g(x) is in the domain of f. In simpler terms, both the inner function and the outer function must be defined for the composition to exist at a particular x-value.
How to Use the Compositions of Functions Calculator
Our calculator simplifies the process of evaluating composite functions at a specific numerical point. Follow these steps:
Enter Function f(x): Type the expression for your first function into the "Function f(x)" field.
Enter Function g(x): Type the expression for your second function into the "Function g(x)" field.
Enter Value of x: Input the numerical value at which you want to evaluate the composite function. This calculator performs numerical evaluation only.
Select Composition Type: Choose whether you want to calculate f(g(x)) or g(f(x)) using the radio buttons.
Click 'Calculate Composition': The result will be displayed below the button.
Important Notes for Inputting Expressions:
Multiplication: Always use an asterisk (*) for multiplication (e.g., 2*x, not 2x).
Powers: Use double asterisks (**) for powers (e.g., x**2 for x squared) or Math.pow(base, exponent).
Mathematical Functions: Prefix common mathematical functions with Math. (e.g., Math.sin(x), Math.cos(x), Math.tan(x), Math.sqrt(x) for square root, Math.log(x) for natural logarithm, Math.abs(x) for absolute value).
Constants: Use Math.PI for π and Math.E for Euler's number.
Parentheses: Use parentheses to ensure correct order of operations.
Examples of Function Composition
Example 1: Basic Polynomials
Let f(x) = x^2 + 1 and g(x) = 2x - 3. Evaluate at x = 2.
To calculate f(g(2)):
First, find g(2) = 2*(2) - 3 = 4 - 3 = 1.
Then, find f(1) = (1)**2 + 1 = 1 + 1 = 2.
So, f(g(2)) = 2.
To calculate g(f(2)):
First, find f(2) = (2)**2 + 1 = 4 + 1 = 5.
Then, find g(5) = 2*(5) - 3 = 10 - 3 = 7.
So, g(f(2)) = 7.
Using the calculator: Enter x**2 + 1 for f(x), 2*x - 3 for g(x), and 2 for x. Select f(g(x)) to get 2, or g(f(x)) to get 7.
Example 2: Involving Square Roots
Let f(x) = Math.sqrt(x) and g(x) = x + 5. Evaluate at x = 4.
To calculate f(g(4)):
First, find g(4) = 4 + 5 = 9.
Then, find f(9) = Math.sqrt(9) = 3.
So, f(g(4)) = 3.
Using the calculator: Enter Math.sqrt(x) for f(x), x + 5 for g(x), and 4 for x. Select f(g(x)) to get 3.
Example 3: Rational Functions
Let f(x) = 1/x and g(x) = x - 1. Evaluate at x = 2.
To calculate g(f(2)):
First, find f(2) = 1/2 = 0.5.
Then, find g(0.5) = 0.5 - 1 = -0.5.
So, g(f(2)) = -0.5.
Using the calculator: Enter 1/x for f(x), x - 1 for g(x), and 2 for x. Select g(f(x)) to get -0.5.