Can a Function be Bijective?

📌 CONCEPT: A function can be bijective if it is both one-to-one (injective) and onto (surjective), meaning each element in the domain maps to a unique element in the range and every element in the range is mapped to by at least one element in the domain.

📐 RULE / FORMULA: For a function to be bijective, it must satisfy the conditions of both injectivity and surjectivity. This can be checked using the horizontal line test, where a function is injective if no two distinct elements in the domain map to the same element in the range, and surjective if every element in the range is mapped to by at least one element in the domain.

💡 WORKED EXAMPLE: Consider the function f(x) = x^3 - 6x^2 + 11x + 2. To check if it's bijective, we first check if it's injective by seeing if any two distinct elements in the domain map to the same element in the range. Then, we check if it's surjective by seeing if every element in the range is mapped to by at least one element in the domain. By analyzing the function's graph, we can see if it passes the horizontal line test.

⚠️ COMMON MISTAKE: Students often mistake a function being bijective with it being both injective and surjective. However, being bijective means that a function is both injective and surjective, which is a stronger condition than just being either one or the other.