Many of the things that happen around us are related to mathematics and if we look closely, we can see that nature itself is related to science; in many physical phenomena there are quantities that define other quantities. For example, how many people can sit on a piece of furniture? That will depend on another quantity that will be the size of the furniture.
We can say that functions they are a set of rules that assign elements to other elements. A set we will call it domain, if an element is taken from this set and we apply a function, it automatically assigns another element of a set that we will call rank.
We can say that a function is a set of ordered pairs (x, y) (domain, range), where the first element is never repeated.
Observing Figure I, in an analogous way, we treat the function as a machine, where we grab a number from the domain set, the function is applied to it and it throws us another number from the range set. Following the drawing, let us observe that we take domain 4, it gives us 16, if we grab 5, it gives us 25, if it is 7, it throws 49, and if it is 2, it throws 4. What is the function of this machine? The function it has is x², let's see:
f (x) = x²
Let's run this machine:
f (2) = 2² = 2 · 2 = 4
f (5) = 5² = 5 · 5 = 25
f (7) = 7² = 7 · 7 = 49
f (4) = 4² = 4 · 4 = 16
It is important to know that f (x) = y, because when we are in a Cartesian coordinate system, the domain set will be the x axis and the range set is the y axis:
So saying y (x) = x² is fine:
and (2) = 4
and (4) = 16
and (5) = 25
and (7) = 49
By having these numbers we can have the following ordered pairs (x, y) (domain, range), which are points on a graph:
And this is the graphical form of our function: