In mathematics, a composition function is a way of combining two functions where the output of one function becomes the input of another. If you have two functions, f and g, the composition of f with g, denoted as (f o g) or f(g(x)), means you first apply function g to an input value x, and then you apply function f to the result of g(x).
The notation (f o g)(x) is read as "f composed with g of x" or "f of g of x". It's crucial to note that the order matters: (f o g)(x) is generally NOT the same as (g o f)(x).
How it Works:
Input: Start with an input value, x.
Inner Function: Apply the inner function (the second one in the notation, g in f(g(x))) to x. This gives you g(x).
Outer Function: Take the result, g(x), and use it as the input for the outer function (the first one in the notation, f in f(g(x))). This gives you f(g(x)).
Example:
Let's say we have:
f(x) = 2x + 3
g(x) = x^2 - 1
To find (f o g)(x), we substitute g(x) into f(x):
(f o g)(x) = f(g(x)) = f(x^2 - 1)
Now, we replace every x in the expression for f(x) with the expression (x^2 - 1):
function calculateComposition() {
var func1Str = document.getElementById("func1Input").value;
var func2Str = document.getElementById("func2Input").value;
var valueStr = document.getElementById("valueInput").value;
var resultDiv = document.getElementById("result");
resultDiv.textContent = "Calculating…";
try {
var x = parseFloat(valueStr);
if (isNaN(x)) {
throw new Error("Please enter a valid number for 'x'.");
}
// — Function Evaluation Logic —
// This is a simplified evaluator. For complex functions, a more robust parser/evaluator might be needed.
// We'll focus on basic arithmetic and powers.
var evaluateFunction = function(funcExpr, val) {
// Replace 'x' with the value
var expression = funcExpr.replace(/x/g, '(' + val + ')');
// Handle common math functions and operations
expression = expression.replace(/Math.pow/g, 'Math.pow'); // Ensure Math.pow is recognized
expression = expression.replace(/sin/g, 'Math.sin');
expression = expression.replace(/cos/g, 'Math.cos');
expression = expression.replace(/tan/g, 'Math.tan');
expression = expression.replace(/sqrt/g, 'Math.sqrt');
expression = expression.replace(/log/g, 'Math.log'); // Natural log by default
expression = expression.replace(/\^/g, '**'); // Support for caret operator for power
// Use Function constructor for evaluation (use with caution for untrusted input)
var evaluator = new Function('return ' + expression);
return evaluator();
};
// 1. Evaluate the inner function g(x)
var g_x = evaluateFunction(func2Str, x);
if (isNaN(g_x) || !isFinite(g_x)) {
throw new Error("Evaluation of inner function g(x) resulted in an invalid number.");
}
// 2. Evaluate the outer function f using the result of g(x)
var f_g_x = evaluateFunction(func1Str, g_x);
if (isNaN(f_g_x) || !isFinite(f_g_x)) {
throw new Error("Evaluation of outer function f(g(x)) resulted in an invalid number.");
}
resultDiv.textContent = f_g_x.toString();
} catch (error) {
resultDiv.textContent = "Error: " + error.message;
}
}