1. Introduction

The study of algebraic functions begins by introducing variables: symbols that represent where a number can go. Traditionally, the letters x, y, z are used to represent real numbers, while in other contexts, z is preferred for complex numbers. It is also customary to use subscripts when there are many variables. Thus, x_1, x_2, \cdots , x_n are also examples of variables.

Algebraic functions are fundamental in various areas of mathematics and their applications. These functions are defined by algebraic expressions involving basic operations such as addition, subtraction, multiplication, division, powers, and roots of variables. Understanding algebraic functions is essential for studying many branches of pure and applied mathematics, including algebra, calculus, geometry, and number theory. Additionally, they have crucial importance in physics, engineering, economics, and social sciences, as they allow modeling and analyzing real phenomena accurately and efficiently.

In the educational field, algebraic functions serve as a solid foundation for developing abstract thinking and problem-solving skills. Through the study of these functions, students learn to manipulate algebraic expressions and understand the relationships between variables, which is fundamental for advancing in more complex mathematics.

In everyday life, algebraic functions are used in a variety of practical contexts. For example, they are applied in financial management to calculate interest and amortizations, in computer science to develop algorithms, and in engineering to design structures and systems. Algebraic functions are also essential in data analysis and statistical modeling, helping to interpret and predict behaviors based on observed data.

In summary, the study of algebraic functions is not only a cornerstone of mathematics but also has a wide range of practical applications that highlight their relevance and utility in the modern world. With a solid understanding of these functions, complex problems can be addressed, and innovative solutions can be developed in various fields.

2. What are Algebraic Functions?

Algebraic functions are a special type of function. A function is a rule of correspondence between two sets that we represent through the notation:

f: A\longmapsto B

Where A is the input set and B is the output set.

Every function f also has a domain (Dom(f)) and a range (Rec(f)). The domain is the set of all input values for which the function produces a valid result, and the range is the set of all possible outputs of the function. The range is also called the Image, and the domain is called the preimage. When defining a function, it is sometimes customary to write it in either of the following two ways:

f: Dom(f)\subseteq A\longmapsto Rec(f)\subseteq B

f: Dom(f)\longmapsto Rec(f)

Thus, algebraic functions are those written in terms of the algebraic operations of their variables, such as addition, subtraction, multiplication, division, power, and principal root. Additionally, a function is said to be of a real variable if its variables are expected to be replaced by real numbers, of a complex variable if they are expected to be replaced by complex numbers, and so on with any other numerical set. There are also functions of one, two, three, or multiple variables, depending on whether they have one, two, three, or many variables.

2.1. Examples of Algebraic Functions

1. Consider the following function

   \begin{matrix} f : & \mathbb{R} & \longrightarrow & \mathbb{R} \\ & x & \longmapsto & f(x) =x^3 + 5x + \displaystyle \frac{6}{\sqrt{x}} \\ \end{matrix}

   This is an algebraic function of one real variable. Here we can see directly that

   Dom(f) = \{x\in\mathbb{R}\;|\; x\gt 0\} = ]0, +\infty[

   This is because there are no divisions by zero and because the principal root is only well-defined for positive real numbers.

2. Now let’s review the following function

   \begin{matrix} f : & \mathbb{R}^2 & \longrightarrow & \mathbb{R} \\ & (x,y) & \longmapsto & f(x,y) =\displaystyle \frac{2xy + \frac{3}{x^2}}{\sqrt[3]{y-1}} \\ \end{matrix}

   This is a function of two real variables that yields a real number as a result. This is also known as a scalar field. This type of function is beyond the scope of this course, but it is very useful in physics to describe quantities such as temperature or density distributions. The domain of this function can also be seen “at a glance.”

   Dom(f) = \{(x,y)\in\mathbb{R}^2\;|\; x \neq 0 \wedge y\neq 1 \}

2.2. Comments on the Graph and Range

Determining the range is generally complicated. Later, we will see techniques that will allow us to do this easily, even in cases where it would seem impossible to achieve algebraically. However, even with these methods, there will be problems because sometimes techniques beyond the scope of this course are needed, such as the calculation of critical points for identifying maxima and minima in differential calculus. Nonetheless, even without calculus, there is much that can be done, and these things will be covered in due course.

If you are still interested in knowing the range and graph of these functions, you can always turn to Wolfram Alpha. Go to https://www.wolframalpha.com/ and try copying and pasting this:

x^3 + 5x + \dfrac{6}{\sqrt{x}}

to get an idea of how the first example would look. For the second one, copy and paste this:

\dfrac{2xy + \dfrac{3}{x^2}}{\sqrt[3]{y-1}}

3. Other Types of Functions

The functions we study in this course can be divided into two kinds: Algebraic Functions and Transcendental Functions. Algebraic functions, as we have seen, are those that are written in terms of fundamental operations, while transcendental functions cannot be written this way or require expressions composed of infinite operations. Algebraic functions can be further divided into two kinds: polynomial and non-polynomial functions. A polynomial function is any function that can be written as the sum or difference of powers. Something like this:

\displaystyle P(x) = \sum_{i=0}^n a_i x^i = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n

Any function that is not of this form is non-polynomial. Among the non-polynomial functions, rational functions stand out, which are those that can be written as the quotient between two polynomial functions.