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Mathematics · 200 Level
This is the complete lecture note for MTH 201 Mathematical Methods I, prepared for Mathematics students at 200 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1 Real-Valued Functions of a Real Variable Series outline Part 1.1: Concept of Real-Valued Functions Definition of a real-valued function Domain, codomain, and range Graphical representation of functions Part 1.2: Types and Properties of Real-Valued Functions Polynomial, rational, trigonometric, exponential, and logarithmic functions Properties: continuity, monotonicity, boundedness Symmetry and periodicity Part 1.3: Operations on Real-Valued Functions Basic operations: addition, subtraction, multiplication, division Composition of functions Inverse functions Part 1.4: Limits and Continuity of Functions Concept of limits Techniques for evaluating limits Continuity and discontinuity Introduction Functions are one of the most important building blocks in mathematics. They describe how one quantity depends on another, and they appear everywhere in science, engineering, and everyday life. For example, when you calculate how distance changes with time, or how profit varies with sales, you are working with functions. In this Series, we focus on real-valued functions of a real variable, which form the foundation for calculus and analysis. This knowledge is crucial
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS because it prepares you for later topics such as differentiation, integration, and multivariable calculus. The aim of this Series is to introduce you to the fundamental concepts, types, and properties of real-valued functions of a real variable, and to equip you with the skills needed to analyse, manipulate, and apply them in mathematical problem solving. We shall commence this Series by examining the concept of real-valued functions, including their definition, domain, codomain, and graphical representation. Next, we will explore the different types and properties of functions, such as polynomial, trigonometric, exponential, and logarithmic forms, as well as continuity and symmetry. Following that, we will consider operations on functions, including composition and inverses. Finally, we will wrap up by studying limits and continuity, which provide the gateway to calculus and deeper mathematical analysis. Series Learning Outcomes Upon completing this Series, you should be able to: SLO 1.1 define real-valued functions of a real variable and identify their domain, codomain, and range, SLO 1.2 explain the main types of real-valued functions, including polynomial, rational, trigonometric, exponential, and logarithmic forms, SLO 1.3 analyse the properties of real-valued functions, such as continuity, monotonicity, boundedness, symmetry, and periodicity, SLO 1.4 demonstrate how to perform operations on functions, including addition, subtraction, multiplication, division, composition, and finding inverses, SLO 1.5 evaluate limits of real-valued functions and assess their continuity and discontinuity. 1.1 Concept of Real-Valued Functions 1.1.1 Definition of a real-valued function A real-valued function is a rule that assigns each element in a set called the domain to exactly one real number in a set called the codomain. The actual set of
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS outputs produced is called the range. For example, the function f(x)=x2takes any real number xand maps it to its square. Fill in the blank: A real-valued function assigns each element in the domain to exactly one __________ in the codomain. Mapping of a real-valued function from domain to codomain 1.1.2 Domain, codomain, and range
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS The domain of a function is the set of input values for which the function is defined. The codomain is the set of all possible outputs, while the range is the actual set of outputs produced. For instance, for f(x)=x, the domain is all non-negative real numbers, the codomain is the set of real numbers, and the range is all non-negative real numbers. Fill in the blank: The set of all actual outputs of a function is called the __________. Table showing definitions of domain, codomain, and range with analogies. Term Definition Vending Machine Analogy Domain The set of all possible input values (usually x) for which the function is defined. The Buttons: The set of all valid codes (like A1, B2) that you are allowed to press. Codomain The set of all potential output values that could theoretically come out of the function. The Inventory: Every item currently sitting inside the glass—all the chips, sodas, and candies. Range The set of actual output values (y) that the function specifically maps to from the domain. The Selection: Only the specific items that will actually fall into the bin when a valid button is pressed. 1.1.3 Graphical representation of functions Functions can be represented visually using graphs. A graph of a function is the set of points x,f(x)plotted in the Cartesian plane. Graphs help us to understand the behaviour of functions, such as whether they increase, decrease, or remain constant. For example, the graph of f(x)=x2is a parabola opening upwards. Fill in the blank: The graph of a function is the set of points (x, __________) plotted in the Cartesian plane. [Insert figure here] Short description: Graph of the function f(x)=x2. Caption: Figure 1.2: Graphical representation of the function f(x)=x2AI prompt: “Generate a graph of the function f(x) = x^2 showing a parabola opening upwards.” Search string: “graph of f(x) = x^2 parabola OER”
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Pilot Questions 1.1 A real-valued function assigns each element in the domain to exactly one: A. Real number in the codomain B. Random value C. Undefined property D. Arbitrary guess (Linked to SLO 1.1) The set of input values for which a function is defined is called the: A. Domain B. Codomain C. Range D. Arbitrary set (Linked to SLO 1.1) The set of all actual outputs of a function is called the: A. Range B. Domain C. Codomain D. Arbitrary guess (Linked to SLO 1.1) The graph of a function is the set of points: A. x,f(x)plotted in the Cartesian plane B. Random points only C. Undefined values D. Arbitrary guesses (Linked to SLO 1.1) The function f(x)=x2has a graph shaped as a: A. Parabola opening upwards B. Straight line C. Circle D. Arbitrary shape (Linked to SLO 1.1) 1.2 Types and Properties of Real-Valued Functions 1.2.1 Types of real-valued functions There are several important categories of real-valued functions, each with distinctive features. A polynomial function is expressed as a finite sum of powers of xwith constant coefficients, for example f(x)=3x2+2x+1. A rational function is defined as the ratio of two polynomials, such as f(x)=x2+1x-2. A trigonometric function involves angles and periodic behaviour, for example f(x)=sinx. An exponential function has the variable in the exponent, such as f(x)=ex. A logarithmic function is the inverse of an exponential function, for example f(x)=lnx. Fill in the blank: A function expressed as the ratio of two polynomials is called a __________ function. Types of real-valued functions with examples
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.2.2 Properties of real-valued functions Functions can be studied through their properties, which describe how they behave. Continuity: A function is continuous if its graph can be drawn without lifting the pen. For example, f(x)=x2is continuous everywhere. Monotonicity: A function is monotonic if it is always increasing or always decreasing. For instance, f(x)=exis always increasing. Boundedness: A function is bounded if its values stay within a fixed range. For example, f(x)=sinxis bounded between -1 and 1. Fill in the blank: A function whose values remain within a fixed range is said to be __________. : Graphical examples of continuity, monotonicity, and boundedness
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1.2.3 Symmetry and periodicity Two further important properties are symmetry and periodicity. An even function satisfies f(-x)=f(x), showing symmetry about the y-axis. For example, f(x)=x2. An odd function satisfies f(-x)=-f(x), showing symmetry about the origin. For example, f(x)=x3. A periodic function repeats its values at regular intervals. For instance, f(x)=sinxhas a period of 2π. Fill in the blank: A function that satisfies f(-x)=f(x)is called an __________ function. Box summarising definitions of even, odd, and periodic functions with examples.
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