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.

[OAU, Ife] MTH 201: Mathematical Methods I

This learning track delivers the complete mathematical toolkit required for a university-level science, engineering, or computing degree. It systematically covers the entire MTH 201 curriculum, building from the foundational principles of single-variable calculus - functions, limits, continuity, and differentiability - to the advanced methods of multivariable calculus, infinite series, numerical methods, and ordinary differential equations. This is the definitive preparation for advanced quantitative study. This programme is designed for second-year students offering MTH 201 at Obafemi Awolowo University, Ile-Ife, Nigeria. It is also helpful for any student in a STEM field - including physics, engineering, and computer science - who requires a rigorous and comprehensive command of calculus and its applications. This track delivers a full skill set in mathematical analysis and applied problem-solving. Graduates will be able to solve a wide range of problems, from optimising multivariable functions to modelling dynamic systems with differential equations and testing the convergence of infinite series. This programme directly prepares students for success in advanced courses in vector calculus, partial differential equations, and real analysis, providing the necessary foundation for a career in engineering, data science, or theoretical physics.

[NUC Core] GET 209: Engineering Mathematics I

Master the mathematical language of engineering. This programme delivers the complete analytical toolkit required for a successful engineering career, covering single-variable calculus, multivariable calculus, linear algebra, and vector analysis. It provides the essential foundation for all subsequent engineering courses. This programme is for second-year undergraduate students across all engineering disciplines. It delivers the official NUC CCMAS curriculum for Engineering Mathematics, providing the core training required for advanced modules in mechanics, thermodynamics, and circuit theory. Model and analyse complex physical systems using calculus, linear algebra, and vector analysis. You will be equipped to solve problems in dynamics, statics, and field theory, providing the quantitative proficiency required for advanced engineering study and professional practice.

[UI, Ibadan] MAT 223: Real Analysis

This learning track provides the complete theoretical machinery of single-variable calculus and analysis. We build the subject from first principles, establishing the rigorous logical framework required for advanced quantitative disciplines. This is the 'why' behind the mathematics that powers science and engineering. This track is built for second-year engineering and physical science students, particularly those at the University Of Ibadan. It is also structured for any student requiring the same rigorous theoretical foundation for advanced quantitative study. On completion, you will command the complete theoretical basis of single-variable calculus. You will construct formal proofs, rigorously analyse function behaviour, and determine the convergence of infinite series. This programme provides the non-negotiable prerequisite knowledge for advanced study in differential equations, complex analysis, and theoretical physics.

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