2.2. Types of Functions

A function is one-to-one (injective) if different elements of the domain are mapped to different elements in the co-domain. Example: If 𝑓 = {(1,𝑎),(2,𝑏),(3,𝑐)}, the function is injective because each element in 𝐴 maps to a unique element in 𝐵.

A one-to-one function, also known as an injective function, ensures that no two different inputs from the domain produce the same output in the co-domain. This means that each element in the input set corresponds to a distinct element in the output set. This characteristic allows us to uniquely identify outputs based on their inputs. For instance, if you think of people and their unique phone numbers, where no two individuals can share the same phone number, this illustrates a one-to-one function.

Imagine a classroom where each student has a unique locker. If Student A has Locker 1, and Student B has Locker 2, these locker numbers are unique to each student. This is like a one-to-one function: each student (input) has a distinct locker number (output), ensuring no lockers are shared.

A function is onto (surjective) if every element of the co-domain is mapped to by at least one element from the domain. Example: If 𝑓 = {(1,𝑎),(2,𝑏),(3,𝑏)}, the function is surjective because every element of 𝐵 has at least one corresponding element in 𝐴.

An onto function, or surjective function, means that every possible output in the co-domain is associated with at least one input from the domain. This characteristic assures we can reach every potential output, as nothing in the co-domain is left out. If there’s even one element in the co-domain without a corresponding element from the domain, the function cannot be classified as onto.

Think of a pizza restaurant with a limited selection of toppings. If every type of pizza on the menu (co-domain) has at least one customer (domain) who orders it, then we can say that the orders are an onto function. No pizza topping is left unchosen, just as every output matches an input.

A function is bijective if it is both one-to-one (injective) and onto (surjective). Example: If 𝑓 = {(1,𝑎),(2,𝑏),(3,𝑐)}, the function is bijective because it is both injective and surjective.

A bijective function is a perfect pairing: not only does each input connect with one distinct output (injective), but every output is connected to at least one input (surjective). This creates a one-to-one relationship, where no inputs or outputs are left without a match. Such functions are very useful as they allow us to easily reverse the function, creating an inverse.

Think about a matching game where each player is given a unique card, and every card corresponds with a player. If each player holds one card and every card belongs to a player, this exemplifies a bijective function. Here, every player can be matched uniquely with their card, and vice versa, eliminating any possibilities of pairing without a match.

• Domain: The set of all possible input values for a function. • Co-domain: The set of possible output values for the function. • Range: The set of actual output values of the function. For example, in the function 𝑓:𝐴 → 𝐵, the domain is 𝐴, the co-domain is 𝐵, and the range is the set of values that 𝑓 maps to in 𝐵.

In a function, different terminologies are used to describe various sets of values. The domain refers to all the potential input values you can use in the function. The co-domain is a broader category that includes all possible outputs, while the range is the actual set of outputs produced by the function based on the inputs from the domain. This differentiation helps in understanding how functions operate and connect between input and output.

Consider a vending machine: the coins you put in represent the domain (input values), the selection of drinks it can provide represents the co-domain (all possible outputs), and the actual drink you receive when you press a button based on the coins you inserted is analogous to the range (the actual outputs that occur).

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