Function Composition Dilemma?

📌 CONCEPT: The concept of function composition states that we can find the composite function f(g(x)) by substituting the expression of g(x) into f(x), where f(x) and g(x) are two given functions.

📐 RULE / FORMULA: To find f(g(x)), we substitute g(x) into f(x) in place of x, giving us f(g(x)) = f(g(x)) = f(x^2 - 4) for our given functions f(x) and g(x).

💡 WORKED EXAMPLE: We can find f(g(x)) for the given functions f(x) = 2x + 1 and g(x) = x^2 - 4. Substituting g(x) into f(x), we get f(g(x)) = 2(x^2 - 4) + 1 = 2x^2 - 7. Now, to find a value of x such that f(g(x)) equals 2x + 1, we set the two expressions equal to each other and solve for x.

⚠️ COMMON MISTAKE: Students often assume that the composite function f(g(x)) will always be equal to the original function f(x), which is not true in all cases. In this problem, f(g(x)) = 2x^2 - 7, which is not equal to 2x + 1.