Function Composition: A Hidden Pattern?

📌 CONCEPT: The composition of functions, denoted as h(x) = f(g(x)), represents a new function that is a result of applying one function to the output of another function.

📐 RULE / FORMULA: The composite function h(x) = f(g(x)) is obtained by replacing the input variable x in the function f with the expression g(x), and then simplifying the resulting expression.

💡 WORKED EXAMPLE: Let's consider two functions, f(x) = 2x^2 and g(x) = 3x + 1. To find the composite function h(x) = f(g(x)), we first substitute g(x) into f(x): h(x) = f(g(x)) = f(3x + 1) = 2(3x + 1)^2. Simplifying further, we get h(x) = 2(9x^2 + 6x + 1) = 18x^2 + 12x + 2. This composite function represents a quadratic function with a coefficient of 18, indicating a greater spread compared to the original function f(x).

⚠️ COMMON MISTAKE: Students often confuse the order of function composition, applying the functions in the wrong order, which leads to an incorrect composite function.