A **function** is a special type of relation between two sets, where each element in the first set (known as the **domain**) is paired with exactly one element in the second set (called the **codomain**). The rule that describes how the elements in the domain are paired with elements in the codomain is called the **mapping rule**. Let’s take a moment to explore how we define a function and understand how we can determine whether a given mapping rule truly qualifies as a function.

To decide whether a given mapping rule is indeed a function, we need to ensure that for every element in the domain, there is **one and only one** corresponding element in the codomain. This means that no element in the domain can be mapped to more than one element in the codomain. If even one element in the domain is paired with multiple elements in the codomain, then the mapping rule does not define a function.

In our exploration, we’ll look at different types of relations and use practical examples to determine whether they meet this requirement. By carefully examining how each element of the domain maps to the codomain, we can develop a clear understanding of what makes a rule a valid function and what distinguishes it from other types of mappings. This understanding is crucial as functions form the foundation for many mathematical concepts and applications.

## What is Image of X under function F? What is Source of Y under F?

In the next tutorial, we’ll dive into the key concepts of **images** and **sources** in the context of functions. These ideas play a crucial role in understanding how functions operate and how they relate different sets of values.

First, let’s tackle the question: **What is the image of XXX under a function FFF?** The image refers to the set of all output values that the function FFF assigns to the elements of XXX. In simpler terms, if you take an input from set XXX and apply the function FFF, the image is the result of that operation. We’ll explore how to find the image of a given set and understand why it’s so important in mapping elements between different sets. This concept helps us visualize what values are “reachable” through the function, giving us insight into the behavior of FFF.

Next, we’ll shift our focus to understanding the **source of YYY under a function FFF**. The source (also called the preimage or inverse image) of YYY under FFF is the set of all elements in the domain of FFF that are mapped to elements in YYY. In other words, the source tells us which inputs from the domain lead to specific outputs in the codomain. This concept allows us to work backwards from an output to discover the inputs that generated it, offering a deeper perspective on the relationship between the two sets.

Through detailed examples and interactive explanations, we’ll explore how to calculate both the image and the source for different functions, and why these concepts are so vital in the study of functions. By the end of the tutorial, you’ll not only understand how to find images and sources but also how they help us analyze the structure and properties of functions. So, get ready to sharpen your understanding and see functions in a whole new light!