2.1. Definition of a Function

A function is a special type of relation where each element of the domain (set 𝐴) is related to exactly one element of the co-domain (set 𝐵).

A function can be viewed as a rule or a mapping that takes every input from one set (the domain) and pairs it with a specific output in another set (the co-domain). This means that for every input value, there should be only one output value. This relationship is unique for functions compared to other types of relations where one input could have multiple outputs.

Imagine a soda vending machine where you insert money (input) to get a specific soda (output). If you choose to purchase a cola, the machine will always give you a cola for that input. This is similar to how a function works—one type of input leads to one specific output.

If 𝑓 is a function from set 𝐴 to set 𝐵, then we write it as: 𝑓:𝐴 → 𝐵.

In mathematics, functions have specific notations that indicate their relationship. When we denote a function as 𝑓:𝐴 → 𝐵, we are declaring that the function 𝑓 takes elements from set 𝐴 and maps them uniquely to elements in set 𝐵. This notation helps in understanding where the function begins (the domain) and where it leads to (the co-domain).

Think of it like a mailing address. When you want to send a letter, you write down who the letter is coming from (your address) and where it is going (the recipient's address). Similarly, the function notation tells us the origin of the input and where it goes.

A function associates each element 𝑥 ∈ 𝐴 to a unique element 𝑦 ∈ 𝐵, denoted as 𝑓(𝑥) = 𝑦.

When we say that a function maps an element 𝑥 from the domain 𝐴 to an element 𝑦 in the co-domain 𝐵, we are emphasizing the relationship where every input in 𝐴 has one and only one output in 𝐵. The notation 𝑓(𝑥) = 𝑦 simply tells us that the function 𝑓 takes 𝑥 and produces 𝑦 as a result.

Imagine a simple math operation like doubling a number. If you input 2 into the function f(x) = 2x, the output will be f(2) = 4. No matter what number you input, the function will give you exactly one distinct output.

Example: If 𝐴 = {1,2,3} and 𝐵 = {𝑎,𝑏,𝑐}, a function 𝑓 might be: 𝑓 = {(1,𝑎),(2,𝑏),(3,𝑐)}.

In this example, we define a specific function 𝑓 that maps numbers from set 𝐴 to letters in set 𝐵. Here, the number 1 is related to the letter '𝑎', 2 is related to '𝑏', and 3 is related to '𝑐'. This illustrates how a function behaves by providing unique outputs corresponding to each input in its defined set.

Think of it like a small restaurant's menu. If you order a burger (1), you'll get a plate with 'A' (the burger itself). If you ask for pasta (2), you receive a dish labeled 'B'. Each order (input) gives you a specific dish (output), just as in the function.

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