Mathematical functions. Introduction and types

13/04/2026

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.

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

Intro
Builder

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!

Giants of science

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

Isaac Newton

Archimedes

287 a.C.

–

212 a.C.

Archimedes established fundamental principles of mechanics and hydrostatics, including the famous Archimedes’ principle of buoyancy in fluids

“Give me a place to stand, and I will move the world”

John von Neumann

1903

–

1957

John von Neumann developed the modern computer architecture, contributed to game theory, quantum mechanics, and statistics, transforming mathematics and computer science

“Truth is, at best, a hypothesis”

Become a giant

Your path to becoming a giant of knowledge begins with these top free courses

Free mode

Mathematics 1 Part 2: Integral calculus, algebra, and applications

Free mode

Mathematics 1 Part 1: Differential Calculus

Free mode

Engineering Calculus and Differential Equations

Free mode

Calculus Applied!

Free mode

Pre-University Calculus

Free mode

Maths Essentials

Free mode

How to Learn Math: For Students

Free mode

Introduction to Algebra

Professional development for Educators

Your path to becoming a giant of knowledge begins with these top free courses

Free mode

How to Learn Online

Free mode

Introduction to Data Wise: A Collaborative Process to Improve Learning & Teaching

Free mode

Support kids’ projects: Programming with Scratch

Free mode

Advancing Learning Through Evidence-Based STEM Teaching

Giants of science

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

Isaac Newton

Leonhard Euler

1707

–

1783

Euler developed much of modern mathematical notation and made key contributions to analysis, topology, mechanics, and graph theory

“Nothing takes place in the universe without mathematics expressing it”

Joseph Fourier

1768

–

1830

Fourier developed the Fourier series, allowing the representation of periodic functions and modeling phenomena like heat, sound, and waves. He transformed applied calculus and mathematical statistics

“The profound analogy between heat and light guides scientific research”

Become a giant

Your path to becoming a giant of knowledge begins with these top free courses

Free mode

Mathematics 1 Part 2: Integral calculus, algebra, and applications

Free mode

Mathematics 1 Part 1: Differential Calculus

Free mode

Engineering Calculus and Differential Equations

Free mode

Calculus Applied!

Free mode

Pre-University Calculus

Free mode

Introduction to Algebra

Free mode

Polynomials, Functions and Graphs

Free mode

How to Learn Math: For Students

Professional development for Educators

Your path to becoming a giant of knowledge begins with these top free courses

Free mode

Teach teens computing: Impact of technology

Free mode

Innovating Instruction: Learning Design in the STEM Classroom

Free mode

HP Digital Skills for Educators – Microsoft 365 Copilot

Free mode

Learn Like a Pro: Science-Based Tools to Become Better at Anything

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.

Mathematical functions. Introduction and types

STEM OnLine mini dictionary

Codomain

Correspondence Rule

Dependent Variable

Domain of Definition

Evaluation of a Function

Image of a Point

Independent Variable

Mathematical Function

Preimage

Range or Image

What are mathematical functions

Main types of mathematical functions

Linear functions

Quadratic functions

Exponential functions

Logarithmic functions

Trigonometric functions

Polynomial functions

STEM OnLine mini dictionary

Codomain

Correspondence Rule

Dependent Variable

Domain of Definition

Evaluation of a Function

Image of a Point

Independent Variable

Mathematical Function

Preimage

Range or Image

Mathematical function simulations

Introduction to mathematical functions

Function builder

Giants of science

Archimedes

John von Neumann

Become a giant

Mathematics 1 Part 2: Integral calculus, algebra, and applications

Mathematics 1 Part 1: Differential Calculus

Engineering Calculus and Differential Equations

Calculus Applied!

Pre-University Calculus

Maths Essentials

How to Learn Math: For Students

Introduction to Algebra

Professional development for Educators

How to Learn Online

Introduction to Data Wise: A Collaborative Process to Improve Learning & Teaching

Support kids’ projects: Programming with Scratch

Advancing Learning Through Evidence-Based STEM Teaching

Giants of science

Leonhard Euler

Joseph Fourier

Become a giant

Mathematics 1 Part 2: Integral calculus, algebra, and applications

Mathematics 1 Part 1: Differential Calculus

Engineering Calculus and Differential Equations

Calculus Applied!

Pre-University Calculus

Introduction to Algebra

Polynomials, Functions and Graphs

How to Learn Math: For Students

Professional development for Educators

Teach teens computing: Impact of technology

Innovating Instruction: Learning Design in the STEM Classroom

HP Digital Skills for Educators – Microsoft 365 Copilot

Learn Like a Pro: Science-Based Tools to Become Better at Anything

Test your knowledge

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

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

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?

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

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

You may also be interested

Tools

Scientific calculator

Scientific Dictionary

Premio Mejor Plataforma Online para Educación STEM 2026

We recommend these simulations