Function Composition Puzzle?

📌 CONCEPT: A composite function (f ∘ g)(x) is defined as (f ∘ g)(x) = f(g(x)), where f and g are two given functions. In this case, we need to find the range of values for x such that the composite function (f ∘ g)(x) is equivalent to x^(4/3).

📐 RULE / FORMULA: To find the composite function, we need to apply the function f to the output of function g, which is g(x). The composite function (f ∘ g)(x) is then obtained by substituting g(x) into f(x). In this case, we have (f ∘ g)(x) = f(√x) = 2(√x)^2 = 2x^(2/3).

💡 WORKED EXAMPLE: Let's consider the given composite function (f ∘ g)(x) = x^(4/3). We can rewrite this as (f ∘ g)(x) = (f ∘ g)(x) = f(g(x)) = f(√x) = 2(√x)^2 = 2x^(2/3). Now, to find the range of values for x such that the composite function is defined, we need to ensure that the input of the function f, which is g(x), is non-negative. This means that √x ≥ 0, which implies that x ≥ 0.

⚠️ COMMON MISTAKE: Students often get confused between the composition of functions and the multiplication of functions. The composite function (f ∘ g)(x) is not equal to f(x) * g(x), but rather f(g(x)).