Domain of a Composite Function?

📌 CONCEPT: The domain of a composite function (f ∘ g)(x) is the set of all possible values of x in the domain of g(x) such that the output of g(x) is in the domain of f(x).

📐 RULE / FORMULA: To find the domain of (f ∘ g)(x), we need to identify the values of x for which g(x) is defined and then check if the range of g(x) is within the domain of f(x).

💡 WORKED EXAMPLE: Consider the composite function (f ∘ g)(x) = f(g(x)) = 1/√x. The domain of g(x) = √x is all non-negative real numbers. Now, we need to check if the range of g(x), which is all non-negative real numbers, is within the domain of f(x) = 1/x. Since the range of g(x) is all non-negative real numbers and the domain of f(x) is all non-zero real numbers, the domain of (f ∘ g)(x) is all non-negative real numbers except 0.

⚠️ COMMON MISTAKE: Students often forget to check if the range of g(x) is within the domain of f(x), which can lead to incorrect domain of the composite function.