Relations and Functions | Class 12 Mathematics Notes

1.3 Types of Functions

Building upon the concept of functions introduced in Class XI, this section explores different classifications of functions based on their mapping properties. Consider functions f1, f2, f3, and f4 with domain X1 and co-domain X2, X3, etc., illustrated by diagrams. A function f: X → Y is one-one (injective) if distinct elements in X map to distinct elements in Y; formally, if f(x1) = f(x2) implies x1 = x2. If this condition fails, the function is many-one. For example, f1 and f4 are one-one functions, while f2 and f3 are many-one. A function is onto (surjective) if every element of the co-domain Y is an image of some element in X. Functions f3 and f4 are onto, whereas f1 is not onto as some elements in Y have no pre-image. A function that is both one-one and onto is called bijective. The section provides examples such as the roll number function (one-one but not onto), f(x) = 2x on natural numbers (one-one but not onto), and f(x) = 2x on real numbers (bijective). It also discusses functions that are neither one-one nor onto, like f(x) = x² on real numbers. The section highlights a key property of finite sets: a function from a finite set to itself is one-one if and only if it is onto, a property not necessarily true for infinite sets. This section also introduces the concept of identity, constant, polynomial, rational, modulus, and signum functions as special cases studied earlier.

📊 Diagram: (i); (ii); (iii); (iv)

🔗 Connection: Understanding types of functions prepares students for the next section on composition of functions and invertible functions.