Domain Restriction vs. Co-domain Constraint?

📌 CONCEPT: A domain restriction of a function f: A → B is a subset A' of A, whereas a co-domain constraint of a function g: A → B is a subset B' of B. The two are equal if the restricted domain of f equals the co-domain constraint of g, and both functions are bijective.

📐 RULE / FORMULA: The condition for f and g to be equal is given by A' = B' and both f and g being bijective.

💡 WORKED EXAMPLE: Let f: R → R be defined by f(x) = x^2 and g: R → R be defined by g(x) = -x. Suppose we have the domain restriction A' = [0, ∞) for f and the co-domain constraint B' = (-∞, 0] for g. Then, A' = B' and both f and g are bijective, so they are equal.

⚠️ COMMON MISTAKE: Students often confuse the domain restriction and co-domain constraint, and may not consider the bijectivity of the functions.