MTS102 - Session 1 Material PDF

Study Session 1 Study Session 1: Functions of a Real Variable Content Introduction Learning Outcomes for Study Session 1 Learning Activities 1.1 Definitions of Terms 1.2 Functions 1.3 Inverse of a function Summary for Study Session 1 Glossar...

Study Session 1 Study Session 1: Functions of a Real Variable Content Introduction Learning Outcomes for Study Session 1 Learning Activities 1.1 Definitions of Terms 1.2 Functions 1.3 Inverse of a function Summary for Study Session 1 Glossary of Terms Pilot Answers Self-Assessment Questions (SA Qs) for Study Session 1 References/Further Reading 3|P age Study Session 1: Functions of a Real Variable Introduction In mathematical analysis, applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers or a subset of that contains an interval of positive length. Learning Outcomes for Study Session 1 By the end of studying this unit, you should be able to 1.1 Define basic terms such as, function, periodic function, monotonic function, even and odd functions etc 1.2 Identified one-to-one functions, onto function, composite and inverse functions 1.3 Solve simple problems on functions of real variable 4|P age Unit 1.1 Definition of Terms Function: Let and be real number, such that a relation between and such that is given, if the y is determined, we say that is a function of and is called independent variable and is the dependent variable, that is. E.g: If , then if , respectively Box 1.1: Monotonic Function is monotonic increasing if whenever is monotonic decreasing if whenever. What are periodic, integral, monotonic, even and odd functions? Periodic Function: A function which repeats itself at a regular interval of is called periodic. Integral of Definition: The range of values of for is defined is called integral of definition. E.g if the function is undefined. If or √ ,. Then the integral of definition for this function is The function is defined for. Monotonic Function: is monotonic increasing if whenever is monotonic decreasing if whenever. Even and Odd Function: (i) A function is said to be even if. E.g (ii) A function is said to be Odd if E.g. Pilot Question 1.1 1. What is a function? 2.Explain even and odd function 5|P age Unit 1.2 Functions Given two non-empty sets and , if there is a rule, which assign an element , a unique element , such a rule is called a mapping. A function is a rule for transforming a member of one set A to a unique member of another set B. A function from a set A to set B is a rule which associates with each member of A a unique member of B. Then. A is called the Domain of the function and B is the Co-domain. A subset of the Co-domain, which is a collection of all the images of the elements of the domain is called the Range. Box 1.2: Domain and co-domain of function A function from a set A to set B is a rule which associates with each member of A a unique member of B. Then. A is called the Domain of the function and B is the Co-domain. Some examples on domain and range of functions Example 1: What is the domain and range of the function. Solution: For any real number its square is uniquely defined. Therefore, the domain of is the set. The square of any number is never negative and the square root of any positive real number exists. Therefore, the range is the set of non-negative real numbers. Example 2: Find the range and domain of Solution: The domain is the set { } Therefore, { } The range is the set of real number between 0 and 1, i.e { } One-to-One Function Function for which different inputs always give different outputs are called one-to-one function (injective). Thus is one-to-one, if implies that or implies that. Note: If one input gives two different outputs, then the mapping is not a function. Example: If and { } Diagram: E.g then injective 6|P age  injective not injective is injective is injective Onto Function These are functions whose range is equal to the Codomain (subjective) while the mapping is bijective, if it is both injective and subjective. Composite Function Suppose and are two functions. Then where is the composite function. For example: If and , find , ,. Solution: √ =√ = 5 and √ =√ = 3 and √ √ √ =√. Pilot Question 1.2 What is the domain and range of the function. 7|P age Unit 1.3 Inverse of a function Let. The inverse of if it exists is the function such that for all and if then (invertible function). Some examples on inverse of a function Example1: If Determine the functions Solution: , let For , Let 8|P age Example 2: Let where , find. Solution: , Let Pilot Question 1.3 If where , find. Summary of Study Session 1 9|P age Function: Let and be real number, such that a relation between and such that is given, if the y is determined, we say that is a function of and is called independent variable and is the dependent variable, that is. Given two non-empty sets and , if there is a rule, which assign an element , a unique element , such a rule is called a mapping. A function is a rule for transforming a member of one set A to a unique member of another set B. A function from a set A to set B is a rule which associates with each member of A a unique member of B. Then. A is called the Domain of the function and B is the Co-domain. A subset of the Co-domain, which is a collection of all the images of the elements of the domain is called the Range. 10 | P a g e GLOSSARY OF TERMS Periodic Function: A function which repeats itself at a regular interval of is called periodic. Integral of Definition: The range of values of for is defined is called integral of definition. Composite Function Suppose and are two functions. Then where is the composite function. 11 | P a g e Self-Assessment Questions (SAQs) for Study Session 1 This unit introduced the students to the basic knowledge of functions in mathematics. It defined some terms and solve some problems on functions. 1. If and where and find (i) (ii) (iii) and (iv) 2. If , and , find , stating the value of for which is not defined. Hence, find. 3. The operation is defined over by Find. 4. State the domain and range of the following (i) (ii). 12 | P a g e References/Further Reading Introductory Mathematics II-futanewsandgist. University: Federal University of Technology Akure. http://www.studocu.com Introductory Mathematics for tertiary institutions http://www.researchgate.net Elementary mathematics ii http://nou.edu.ng National Open University of Nigeria 13 | P a g e