Functions and Their Graphs - Single-Variable Calculus (Undergraduate Foundation)

This course provides a rigorous foundation in the concept of functions, the building block of calculus. We begin with the real number system - rationals, irrationals, intervals, and infinity - before defining real-valued functions, their domains, and ranges. You will systematically study all key function families: polynomial, rational, algebraic, piecewise-defined, and transcendental, including exponential, logarithmic, trigonometric, and hyperbolic types. A command of functions is essential for all technical and quantitative disciplines. These principles are directly applied in physics to model motion, in economics for supply-demand curves, in engineering for signal processing, and in computer science to analyse algorithm complexity. This course delivers the core mathematical tools required for effective modelling and problem-solving in these fields and beyond. By the end of this course, you will be able to: calculate the domain and range of any given function; classify functions and identify their fundamental properties; analyse functions for key characteristics like symmetry (odd and even); and accurately construct graphs for a wide range of standard and piecewise-defined functions directly from their equations. This course is designed for first-year undergraduates in science, technology, engineering, mathematics (STEM), and economics. It is also critical for any student preparing for advanced calculus or for professionals who require a solid mathematical base for roles in finance, data science, or technical research.

7 hrs

Payment required for enrolment

Enrolment valid for 12 months

This course is also part of the following learning tracks. You may join a track to gain comprehensive knowledge across related courses.

[FUTA, Akure] MTS 102: Introductory Mathematics II

This learning track is structured for first-year students at the Federal University of technology, Akure (FUTA) and mirrors the standard second-semester coverage of elementary calculus. It begins with single-variable functions and their graphs, then walks learners through limits, continuity, differentiation techniques, and curve sketching—just as covered in the official MTS 102 outline. You’ll also explore anti-derivatives and integration, learning both the techniques and how to apply them to solve practical problems in science and engineering contexts. Everything is broken down into short, focused video lessons that keep things clear and manageable, especially for students who might be engaging this content for the first time. If you're not a FUTA student but need to build a solid foundation in these same topics, this track can serve you just as well. The structure and explanations are universal, ensuring that learners with similar academic goals can benefit fully.

[UNILAG, Akoka] MTH 102: Elementary Mathematics II

This learning track is designed to guide first-year students at the University Of Lagos through key concepts in calculus, beginning with the fundamentals of single-variable functions and their graphs. It builds gradually into the core topics of limits, continuity, and differentiability, with each course tailored to simplify these foundational ideas for early learners. The focus is not just on theory but also on building the skill to solve problems confidently, especially those typically encountered in university-level exams. You’ll move from understanding the concept of a limit to mastering how derivatives work and how to apply them to sketch curves and analyze function behavior. Although built for UNILAG students, this track is suitable for anyone looking to strengthen their understanding of introductory calculus at the university level. Whether you're preparing for school assessments or seeking a solid refresher, this track will help you follow a structured path.

[OAU, Ife] MTH 201: Mathematical Methods I

Comprehensive treatise of advanced calculus covering limits, continuity and differentiability, infinite sequences and series, partial differentiation, numerical methods and ordinary differential equations. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Course Chapters

1. Introduction

This chapter establishes the foundational concepts of the real number system. A command of these definitions is a prerequisite for defining and analysing the functions that form the core of this course. By the end of this chapter, you will be able to: differentiate between key number systems from natural to real; define and represent intervals on the real line; and correctly apply the mathematical concept of infinity.

Chapter lessons

1-1. Number Systems

18:36

This lesson defines the hierarchy of number systems, from natural numbers through to the real numbers. These classifications provide the essential vocabulary for discussing functions and their domains throughout this course.

1-2. Rationals and irrationals

26:08

A closer look at differences between rational and irrational numbers.

1-3. Intervals

16:55

Meaning and examples of intervals on the real line.

1-4. Infinity

14:44

Meaning and use of infinity.

2. Real-Valued Functions

This chapter introduces the function - the central object of study in calculus. We will formally define what constitutes a real-valued function and establish the critical concepts of its valid inputs (domain) and possible outputs (range). By the end of this chapter, you will be able to: formally define a function; precisely calculate the domain for various real-valued functions; and understand the meaning and importance of a function's range.

Chapter lessons

2-1. Introduction

24:36

Meaning of functions and real-valued functions.

2-2. Domain of functions

31:32

Domain of functions and its calculation.

2-3. Range of functions

6:17

Meaning of the range of functions.

3. Kinds of Real-Valued Functions

This chapter moves from the general definition of a function to a systematic classification of key function families. We will identify and analyse the core algebraic types - including polynomial, rational, and piecewise functions - that form the basis of mathematical modelling across scientific disciplines. By the end of this chapter, you will be able to: classify functions as polynomial, rational, or algebraic; determine the natural domain for each of these types; interpret and analyse piecewise-defined functions; and test any function for symmetry.

Chapter lessons

3-1. Polynomials