Mathematical functions. Introduction and types

05/05/2026

The online mathematical function simulations on this page serve as a first analysis and introduction to mathematical functions. In addition, we will see some of the main types of mathematical functions and with the function generator we will create some examples.

This Thematic Unit is part of our Mathematics collection

Codomain

The set of possible values the function could take, within which the range or image is found.

Correspondence Rule

An algebraic expression or algorithm that defines exactly how to transform a domain value into a codomain value.

Dependent Variable

A magnitude whose value is determined by the function’s correspondence rule when applied to the independent variable.

Domain of Definition

The set of all real numbers for which the function is mathematically defined and produces a real result.

Evaluation of a Function

The procedure of substituting the independent variable with a number to find its corresponding value in the function.

Image of a Point

The specific value returned by the function when evaluated at a concrete value within its domain.

Independent Variable

An element of the domain whose value does not depend on another variable and acts as the input argument of the function.

Mathematical Function

A dependency relationship between two magnitudes where each input value corresponds to exactly one output value.

Preimage

The set of domain values that, when processed by the function, result in a specific value within the range.

Range or Image

The subset of values in the codomain that are actually reached by the function when the full domain is applied.

What are mathematical functions

Mathematical functions are fundamental tools in the study of relationships between variables. They are expressions that relate one or more variables and generate a specific output or result. These functions can be represented in various forms, such as algebraic equations, graphs or tables of values.

Main types of mathematical functions

There are many types of mathematical functions, each with distinct characteristics and properties. Some of the main types of mathematical functions are as follows:

Linear functions

Linear functions are those whose graphical representation is a straight line. They have the form f(x) = mx + b, where m is the slope and b is the ordinate to the origin.

Quadratic functions

These are functions of second degree, whose graphical representation is a parabola. They have the form f(x) = ax² + bx + c, where a, b and c are constants.

Exponential functions

They are those in which the independent variable is in the exponent. They have the form f(x) = a^x, where a is a constant and x is the variable.

Logarithmic functions

They are the inverse of the exponential functions. They have the form f(x) = log_ax, where a is a constant and x is the variable.

Trigonometric functions

They include the sine, cosine, tangent functions, among others. These functions are related to the angles of a triangle and have applications in geometry, physics and other disciplines.

Polynomial functions

They are those that are formed by an addition or subtraction of terms of integer powers. They have the form f(x) = a_nxⁿ + a_n-1x^n-1 + … + a₁x + a₀,, where a₀, a₁, …, a_n are constant coefficients.

These are just a few examples of mathematical functions. The choice of the appropriate function depends on the context and the relationship to be modeled. The study and understanding of mathematical functions are fundamental to solve problems and analyze phenomena in various areas of knowledge.

Codomain

The set of possible values the function could take, within which the range or image is found.

Correspondence Rule

An algebraic expression or algorithm that defines exactly how to transform a domain value into a codomain value.

Dependent Variable

A magnitude whose value is determined by the function’s correspondence rule when applied to the independent variable.

Domain of Definition

The set of all real numbers for which the function is mathematically defined and produces a real result.

Evaluation of a Function

The procedure of substituting the independent variable with a number to find its corresponding value in the function.

Image of a Point

The specific value returned by the function when evaluated at a concrete value within its domain.

Independent Variable

An element of the domain whose value does not depend on another variable and acts as the input argument of the function.

Mathematical Function

A dependency relationship between two magnitudes where each input value corresponds to exactly one output value.

Preimage

The set of domain values that, when processed by the function, result in a specific value within the range.

Range or Image

The subset of values in the codomain that are actually reached by the function when the full domain is applied.

Explore the exciting STEM world with our free, online, simulations and accompanying companion courses! With them you’ll be able to experience and learn hands-on. Take this opportunity to immerse yourself in virtual experiences while advancing your education – awaken your scientific curiosity and discover all that the STEM world has to offer!

Mathematical function simulations

Introduction to mathematical functions

The first of our mathematical function simulations will serve as an introduction to mathematical functions. Play with functions while reflecting on the History of Art. Look for patterns, then apply what you’ve learned on the Mystery!!! screen.

Function builder

Play with functions while reflecting on Art History. Explore geometric transformations and change your thinking about linear functions, then have fun discovering some of the main types of mathematical functions!

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“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

1643

–

1727

The supreme genius of the Scientific Revolution who unified the heavens and the Earth under the laws of gravity.

Carl Friedrich Gauss

1777

–

1855

The prince of mathematics who tamed the normal curve of errors and instrumented terrestrial magnetism.

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Isaac Newton

Gottfried Wilhelm Leibniz

1646

–

1716

The universal genius who co-invented infinitesimal calculus and designed the binary system that powers computers.

John von Neumann

1903

–

1957

The brilliant polymath who designed modern computer architecture and founded game theory.

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Test your knowledge

What is a mathematical function, and why is it such a fundamental tool for describing relationships between quantities?

A mathematical function is a rule that assigns each element of a domain to exactly one element of a codomain. This structure allows us to describe how one quantity depends on another, making functions indispensable across mathematics, physics, engineering, economics, and countless applied fields. Functions let us model real‑world phenomena, analyze how systems change, and express complex relationships in a precise and manipulable way. Whether we’re studying population growth, electrical current, chemical reactions, or geometric transformations, functions provide the formal language that connects variables and reveals underlying patterns.

What are the essential components of a function, and how do they help characterize its behavior?

The key components of a function are the domain, the codomain, the assignment rule, and the independent and dependent variables. The domain specifies which inputs are valid; the codomain indicates the type of outputs the function can produce; and the assignment rule defines how each input is mapped to its output. These elements allow us to analyze continuity, monotonicity, symmetry, periodicity, extrema, and many other properties. Graphical representations further enrich this understanding by providing an intuitive view of how the function behaves across its domain.

Why does a function have to give only one output for each input? Wouldn’t it be more flexible if it could return several values?

If a single input produced multiple outputs, the relationship would no longer be predictable or mathematically manageable. The whole point of a function is determinism: one input, one output. That’s what allows us to analyze it, graph it, differentiate it, or use it in models. Relationships that assign several outputs do exist, but they belong to a different category—relations or correspondences—not functions. A function must behave consistently to be useful.

Why do we bother talking about domain and codomain? Isn’t the formula enough to understand the function?

The formula alone doesn’t always tell you where the function makes sense or what kinds of values it can produce. The domain clarifies which inputs are allowed—avoiding undefined operations like dividing by zero or taking square roots of negative numbers in the real system. The codomain tells you the “universe” where the outputs live, which matters when analyzing ranges, limits, or transformations. Without domain and codomain, you only have half the picture.

How come a graph helps so much in understanding a function? Isn’t it just a drawing?

A graph turns an abstract rule into a visual landscape. It lets you instantly see whether the function rises, falls, oscillates, levels off, or has abrupt changes. Features like maxima, minima, asymptotes, discontinuities, and inflection points become obvious at a glance. Many behaviors that are hard to detect from an algebraic expression become crystal clear when plotted. That’s why graphing is one of the most powerful tools for understanding functions in mathematics and science.