Function Composition: Insight?

📌 CONCEPT: Function composition is the process of combining two or more functions to create a new function, where the output of one function serves as the input for the next function.

📐 RULE / FORMULA: To evaluate (f ∘ g)(x), we need to first find the value of g(x) and then substitute it into f(x).

💡 WORKED EXAMPLE: For the given functions f(x) = 2x + 3 and g(x) = 3x − 2, we can find the value of (f ∘ g)(5) as follows: We first find g(5) = 3(5) - 2 = 13. Then, we substitute g(5) into f(x) to get f(g(5)) = f(13) = 2(13) + 3 = 29. Therefore, (f ∘ g)(5) = 29.

⚠️ COMMON MISTAKE: Students often confuse the order of function composition and evaluate (f ∘ g)(x) as f(x) ∘ g(x) instead of f(g(x)).