Final+Version

== =Functions=

Definition:
A Function is a operational rule where a group of objects in a set A, have assigned a one object and only one object in a set B. In mathematics, the group of objects are Real Numbers (lR). Therefore, when we talk about functions, we are assigning one number of set A to only and only one number of set B.

Description:
The Functions are characterized to represent, in algebraic language, set's of numbers A and B. The functions uses the variables to __indentif__y and modify mathematic relations between the sets A and B, using some algebra.

For example, we can express the relation between the set A and set B using the next function:

y = x^2 + 3

Where "x" represent the set A, and "y" the set B. They are called "Variables". On the other hand, "x" is called too, "Independ Variable" and "y" depended variable.

Other notation used to represent a function is the notation "f of x", that we can express it across the next structure:

f(x) = x^2 + 3

Where "f(x)" has the same use like "y", with the diference that it can be tell us, what value was assigned to the variable x.

For Example:

f(3) = (3)^2 + 3 = 12

Classification
The Functions can be classified into three branches: Affine functions, Exponential Functions and Trigonometric Functions.

· ** Affine Functions: ** These functions are those that changes at constant rate with respect to its independent variable. These can be written with the expression:

y = M x + b ; or y = x^n + x^n-1 +...+ x^n-m + b

Where: "**y**" It is the dependent variable. "**x**" It is the independent variable. "**M**" It is the slope of the function. "**b**" It is a constant that represent a court on Y axis by the line expressed by the function in the Euclidean space.


 * Exponential Functions: ** These functions are of the form:

y = a ^ x

"**a**" It is a constant, that represent the **Base** "**x**" it is the Independent variable and represent the **Exponent** of the function. and has a exponential growing with a constant rate respect to the values assigned to its independent variable.

· ** Trigonometric Functions: ** These functions are those that establish a relation between the angles and its oscillatory characteristic, with circumscribed triangles in a circumference, where the independent variable in the function represents the sweep angle.

Some expressions of trigonometric functions are:

y = Sen ( x ) ; y = Cos ( x )

y = Sen (x) / Cos (x) = Tg ( x )

Comparison and Contrast:
The function has features that difference it of the other operations in calculus. When you assign a value to the independent variable, you will obtain a new value of the depended variable. In other words, it works like a machine, where you put the "material" and in the end of the process, you will obtain a "Product".

That is the relevant difference between the functions and other operations in math like the Matrix's, determinants, that doesn't depend of variables.

On the other hand, there are other operations that are functions but uses different operator to obtain some useful values.

For Example: The Integral use functions to calculate the down area of a curve using integral properties to modify the functions.