Domain of a Composite Function?

📌 CONCEPT: The domain of a composite function h(x) = g(f(x)) is the set of all values of x for which the inner function f(x) is defined and the output of f(x) is a valid input for the outer function g(x).

📐 RULE / FORMULA: To find the domain of h(x), we need to consider the restrictions on the domain of f(x) and ensure that g(f(x)) is defined for all values in the domain of f(x).

💡 WORKED EXAMPLE: Let's consider the given functions f(x) = √(x - 4) and g(x) = x^2 + 1. The domain of f(x) is x ≥ 4 because the square root of a negative number is undefined. When g(f(x)) = (f(x))^2 + 1, we need to ensure that f(x) ≥ 0 for g(f(x)) to be defined. Since f(x) ≥ 0 only when x ≥ 4, the domain of h(x) is x ≥ 4.

⚠️ COMMON MISTAKE: Students often forget to consider the restrictions on the domain of the inner function f(x) when finding the domain of the composite function h(x).