Can a function be one-to-many and onto?

📌 CONCEPT: A function cannot be both one-to-many (injective) and onto (surjective) if its codomain is the same as its domain, as the presence of one-to-many (or many-to-one) implies it's not one-to-one, which is a prerequisite for surjectivity in such cases.

📐 RULE / FORMULA: For a function to be injective, it must map distinct elements of the domain to distinct elements of the codomain, whereas for a function to be surjective, every element in the codomain must be the image of at least one element in the domain.

💡 WORKED EXAMPLE: Consider the function f(x) = 2x + 1. It is injective because f(x1) = f(x2) implies 2x1 + 1 = 2x2 + 1, which further implies x1 = x2. However, it is not surjective because not every real number can be obtained by the function, for example, there is no x such that f(x) = -1. Therefore, it cannot be both one-to-many and onto.

⚠️ COMMON MISTAKE: Students often confuse injectivity and surjectivity, assuming that a function can be both one-to-many and onto in certain cases, overlooking the fact that in such scenarios, the codomain should be different from the domain for the function to be onto.