**A function of a real variable is a dependency relationship between a dependent variable (Y) and an independent variable (X). **

In other words, the dependent variable (Y) takes values determined according to (depending) on the values taken by the independent variable (X).

We define:

Independent variable = X = {x _{1, } x _{2} ,…, x _{n} }.

Dependent variable = Y = {and _{1} , and _{2 } ,…, and _{n} }.

The expression “being a function of” can be understood as “being dependent on.” That is, the variable Y is a function of the variable X. The variable Y is called a dependent variable precisely because it depends on the values taken by the independent variable X. In the same way, it is called an independent variable because its value does not depend of any variable expressed in the function.

Generally, for each value of the independent variable X only one single value of the dependent variable Y corresponds. This statement is true as long as we do not take into account other types of functions that allow the dependent variable Y to have more than one value of the associated independent variable X. That is, there are functions where a dependent variable Y, can be related to more than one value of the independent variable X. These types of functions are called surjective functions.

Functions use equations to represent the dependency relationship between the dependent and independent variables. So, the mathematical expression of the equations are the functions. Thanks to the functions, we can represent the equations in the graphs.

**Application of a mathematical function**

In **microeconomics** we use the functions when we want to express the usefulness of the agents that participate in the economy. In **finance** , when we want to express the risk profile of an agent exposed to a situation of uncertainty. In **econometrics** , both linear and nonlinear regressions are also functions.

**Classification of mathematical functions**

The functions can mainly be classified according to their nature and condition:

- Algebraic functions
- Polynomial functions.
- Functions to pieces.
- Rational functions.
- Radical functions.
- Transcendent Functions
- Injecting functions
- Surjective functions.
- Byective functions.
- Non-injective and non-surjective functions.

**Theoretical example**

- Y = 3X.
- The dependent variable Y will be the values taken by the variable X multiplied by 3. The slope of the line is 3 and must pass through the origin of coordinates. The graphic representation is a line.

Graph of a linear mathematical function:

- Y = 4X
^{2}- The dependent variable Y will be the values taken by the variable X squared and multiplied by 4. The graphical representation is a parabola.

Graph of a quadratic mathematical function: