1. My Introduction Name: Prakash ADHIKARI Academic Qualification: Master’s Degree Research, Universite Lumiere-IUT Lumiere, Lyon - 2(ULL-2), France Master’s Degree in Education Majoring Mathematics, TU Nepal B.ED Mathematics - TU, Nepal Teaching Background: Mathematics Instructor, Heartland Academy,Kathmandu 2010 to 2013

2. Relation and Function

3. 1.Cartesian Product 2.Relation 3.Function

4. 4 How many ways each player of one team handshake with players of another team? Germany (A)Argentina (B) Kevin Manuel Lukas Messi Romero zabaleta

5. Cartesian Product 5 A×B = { (kevin,Messi),(Kevin,Romero), (Kevin,zabaleta),(Manuel,Messi ),(Manuel,Romero),(Manuel, Zabaleta),(Lukas,Messi),(Lukas, Romero),(Lukas,Zabaleta) } Kevin Manuel Lukas Messi Romero Zabaleta AB

6. 1 to 1many to many Relations 1 to many

7. Countries UK Nepal India Bangladesh London Dhaka New Delhi Paris Kathmandu Relation Capital Cities France Relation - ‘ is the capital city of’

8. 1 2 3 4 5 2 10 8 6 4 Relation This relation is R= {(1,6), (2,2), (3,4), (4,8), (5,10)} AB

9. This is a relation R={(2,3), (-1,5), (4,-2), (9,9), (0,-6)} Domain = {-1,0,2,4,9} All x Values Range = {-6,-2,3,5,9} All y Values Relations Domain and Range in Relation

10. Function Rice Peeling and Milling Machine f(x) means Function of x

11. Function In a Function, One Input ALWAYS has exactly one output

12. 2 x3x3 8 2 is the input number (or x-value on a graph). 8 is the output number (or y-value on a graph). The illustrates the idea of a function. Function InputOutput x3x3

13. Definition: Function f

14. Then, we say that Set A is the Domain of f Set B is the Co-domain of f If f(a) = b, we say that bєB is the Image of a aєA is the Pre-image of b. The Range of f:A  B is the set of all images of elements of A. Functions: Domain, Codomain and Range 14

15. Consider an example with Set A= { 1,2,3,4} and B={3,6,11,18} We have function f (x) = x 2 +2 from A to B ; Domain of f ={ 1,2,3,4} Then Range of f = {3,6,11,18} Functions: Domain, Codomain and Range 15 : {3,6,11,18} : {1,2,3,4} f(x)= x 2 +2

16. Given f(x) = 4x + 8, find each: 1. f(2) 2. f(a +1) = ? 3. f(  4a) = ? Evaluating Functions = 4(2) + 8 = 16

17. If f(x) = 3x  1, and g(x) = 5x + 3, find each: Evaluating More Functions = ? 1. f(2) + g(3) 2. f(4) - g(-2) 3. 3f(1) + 2g(2)

18. How to Know the given Relation is function or not??? Functions 18

19. Input number Output number Can you have one letter going to two different mail boxes??? Not a FUNCTION To understand in Better way:

20. Input number Output number Can you have two different letters going to one mail box? Can you have a letter going to one mail box?

21. More Ideas.. Voters Candidates 21

22. 1. f is a function 2. 12342341234234 A B 12341234 A B 3 6 11 18 2 3 4 12342341234234 12341234 BABA A BABA

23. f is not a function 3.4. wxyzwxyz 4 A B A B 12342341234234 3 6 11 18 2 3 4 12342341234234 3 6 11 4 A BABA A BABA

24. One-to-one (or injective) Many to One Onto ( Or Surjective) Types of Function 24

25. 1. One to One Function f is one-to-one (or injective) function, if and only if it does not map two distinct elements of A onto the same element of B. In other words: A function f:A  B is said to be one-to-one iff,  x, y  A (f(x) = f(y)  x = y) Types of Function 25 f 12341234 12341234 A B

26. 2. Many to One Function Association of more than one element of Domain with single element in Range. f(1) = 1, f(2) = 1, f(3) = 1 and f(4) = 1 So f is Many-to-one function. Types of Functions 26 f 12341234 1 A B

27. 3. Onto Function: A function f:A  B is called onto (or surjective) function if and only if for every element b  B there is an element a  A with f(a) = b. In other words, If the codomain set is equal to range set then the function is onto or Surjective. Types of Functions 27 Codomain = { 1,4,9) Range of Function = {1,4,9) CODOMAIN = RANGE

28. Types of Function 4. Identity Function Let A be any non- empty set, The function defined by i(a)=a for all aєA, is called Identity Function of Set A Example: Let A= {1,2,3,4} then is given by i(1)=1 i(2)=2 i(3)=3 i(4)=4 28

29. Discussing questions: Define One-to-One Function with example. What is Onto Function? Give an example. 29

30. An inverse function is a Function that "reverses" another function: if the function f applied to an input x gives a result of y under f, then applying its inverse function f -1 to y gives the result x i.e. f(x) = y Iff f -1 (y) = x INVERSE OF FUNCTION 30 y x f f -1 f(x)=y f -1 (y)=x

31. An Example of INVERSE FUNCTION 31 f(a) = 3 f(b) = 1 f(c) = 2 f -1 (3) = a f -1 (1) = b f -1 (2) = c

32. f -1 :C  P is no function, because it is not defined for all elements of C and assigns two images to the pre-image New York. INVERSE OF FUNCTION 32 Linda Max Kathy Peter Boston New York Hong Kong Moscow LübeckHelena f f -1 Inverse of a function MAY NOT BE A FUNCTION

33. Inverse Function 33

34. Composition or Composite Functions 34

35. The composition ( or Composite of two functions) f:A  B and g:B  C, denoted by g 0 f, is defined by [(g 0 f)a] = g(f(a)) Composition 35

36. Example: f(x) = 7x – 4, g(x) = 3x, Function is defined as f:R  R, g:R  R (g o f)(x) = g(f(x)) = g(7x-4) = 3(7x-4) = 21x - 12 If x=5, (g o f)(5) = g(f(5)) = g(31) = 93 Composition 36

37. Composition of a function and its inverse: (f -1  f)(x) = f -1 (f(x)) = x The composition of a function and its inverse is the identity function i(x) = x. Composition 37

38. Given the function f(x)= x-6, xєR, Find the values of : i.f -1 (x) ii. f -1 (12) Given the function f(x)= 4x+9, xєR, Find the values of : i. ff(x) ii. fff(x) Given Functions are f(x)= 4x+9 ; g(x)= x 2 +1, xєR, Find the values of: i. f -1 (x) ii. f o (g)x and iii. g o f(x) 38 Work on paper, Now you try….!

39. Additional Questions Functions f and g are defined by: f:x→2x+3 and g:x→x 2 -6x – Express f -1 (x) in terms of x. – Solve the equation if f(x)=f -1 (x) – Find f -1 g(x) The functions f and g are defined for xєR by f:x→3x+a and g:x→b-2x Where a and b are constants. Given that ff(2)=10 and g -1 (2)=3, -Find the values of a and b. -An expression for fg(x) 39

40. Thank You 40