Domain of a Composite Function?

📌 CONCEPT: The domain of a composite function h(x) = f(g(x)) is the set of all possible input values for g(x), which, when input into g(x), produce valid input values for f(x).

📐 RULE / FORMULA: The domain of h(x) = f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f.

💡 WORKED EXAMPLE: Given f(x) = 0.05x + 5 and g(x) = 2x - 3, we need to find the domain of h(x) = f(g(x)). The domain of g(x) is all real numbers, so we find the domain of h(x) by ensuring g(x) is in the domain of f(x). To do this, we need to ensure g(x) ≥ 0, as the cost function f(x) is defined only for x ≥ 0. Therefore, 2x - 3 ≥ 0, which implies x ≥ 3/2. Hence, the domain of h(x) is [3/2, ∞).

⚠️ COMMON MISTAKE: Students often forget to check the domain of the inner function g(x) before applying the composite function, leading to incorrect or incomplete domain descriptions.