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What is functions?

In today's video, we will discuss a concept that you most likely came across during your high school or middle school math classes. This concept is functions. Believe it or not, few people actually understand what a function truly means. In this somewhat brief video, we'll clear up any confusion around the concept of a function. We will also talk about its notation and the key differences between equations and functions, while pointing out their importance at the end of the video. To make this clearer with an example, imagine we have a function that squares the input and then adds four. Mathematically, the function could be written as: input² + 4. If we input the number 3 into the "box," the output will be 13 (because 3² + 4 = 13). If we input -2, the output will be 8 (because (-2)² + 4 = 8). Another way to understand this is to think of the function as a set of inputs and outputs. The inputs represent the values of x, and the outputs represent the values of y. This can be visually represented by a diagram that shows how each input is connected to its corresponding output. In fact, this is the official definition of a function in mathematics—a function is a relationship that links each element from the input set (x) to only one element in the output set (y). Now, let’s move on to the most important point in this video—the difference between equations and functions. There are two main rules to consider when determining whether an equation represents a function: First Rule: An output (y) can correspond to more than one input (x). Second Rule: An input (x) cannot correspond to more than one output (y). This might sound complicated at first, so let me break it down. Take, for example, the equation y = x² + 4, which, when graphed, forms a curve. Notice that when x = 0, y = 4, and when x = 2 or -2, y = 8. This relationship holds true and follows the rules of a function.