1. Prakash Adhikari Islington college, Kathmandu 1

2. Warm up time.. 1.What is the type of the function alongside? 2.What is the domain of the function? 3.What are ranges of the function? 4.What is the co-domain of the function?

3. Warm Up time.. 1.What is the name of the function g? 2.Tell the domain and range of the function 3.Is it possible to find the inverse of this function, state with reason

4. Discussion

5. Function 5

6. PREVIEW of Last Lecture Definition of Relation and Function Domain and Range of Function Different types of Functions Inverse of Function Composition of two Functions f and g 6

7. Learning description… Various forms of functions One to one Function, Determination of one to one function Inverse of Function (Linear and Quadratic form) Domain and Range of function Graph of the Functions Graph of one to one function and its inverse 7

8. The set of fingerprints is uniquely defined for every person. Set of Islington member Set of fingerprint 8

9. One to One Function Let’s Revise the definition: One to one Function is defined as no two elements in domain of the function has same image in Range In other words, A function f is said to be a one to one function if each element is domain has each image in range. A function f:A→B is an One to One if x=y whenever f(x) = f(y) Remember 9

10. Let's solve an example 10

11. Let's solve an example 11

12. Graphically.., We can also test the given function is one to one or not by Horizontal Line test A function is one to one if a horizontal line intersects(cuts) the graph in only one spot(point) 12

13. Graphs of Functions Graph 1 Graph 2 Graph 3 Graph 4 What about these graphs? Are all these graphs are one to one Function? 13

14. Various forms of Functions 14

15. Graph of Function Work on paper: – Determine whether the functions are one- to- one or not. f:x→ 4x, xєR f:x→ x(x-2), xєR – The function f and g are defined as follows: f:x→x 2 -2x ; xєR Find the set of values of x for which f(x)˃15 State, with a reason, whether f has an inverse or not 15

16. Domain and Range of f: A→B Domain – Domain is all REAL NUMBERS. – Defining Domain: – The set of numbers x for which a function f(x) is defined is called – domain of the function. Range : – Range of f is set of all y values lies in set B – Range is also all REAL NUMBERS – The set of numbers y for which a function is defined as images if every x of domain is called Range of the function 16

17. Domain and Range of f: A→B 17

18. Domain and Range of f: A→B What about quadratic Function?? We have an example: F:x→-x 2 -2x-4 Solution Writing f(x) in the form of a(x+b) 2 +c then y= -(x+1) 2 -3 Or, (x+1) 2 =-3-y Since LHS is perfect square then (x+1) 2 >=0 So as -3-y >=0 It means y<= -3 Range of interval is (-Infinity, -3) 18

19. Domain and Range of f: A→B 19  Find the domain of the following functions and also write in interval notation

20. Inverse of a function It is possible to find the inverse of a function only when the function is one to one. In Inverse function range of f(x) is changed into domain for f -1 (x) Steps to find Inverse for linear function – Suppose f(x) as y – Interchange the value of x and y – Finally, obtain the value of y in terms of x, which is the inverse of Linear Function. 20

21. Inverse of a function 21

22. Methods of finding Inverse of functions 22

23. Inverse of a function Inverse for linear function- f(x)=ax 2 +bx+c Steps: – Suppose f(x) as y – Write the function as y= a(x+b) 2 +c ( by completing the square) – Interchange the value of x and y – Finally, obtain the value of y in terms of x, which is the inverse of Linear Function 23

24. Inverse of Function 24

25. Inverse of Function Work on paper: Find the inverse of following functions 1.f:x→ x 2 -6x, xєR and x>= 0 2.f:x→ x 2 +x+6, xєR and x> 0 3.f:x→ -2x 2 +4x-7, xєR and x<1 4.f:x→ x 2 -2x+7, xєR 25

26. Graph of Function 26

27. Graph basics… Quadrant I X>0, y>0 Quadrant II X 0 Quadrant III X<0, y<0 Quadrant IV X>0, y<0 Origin (0,0) 27

28. Graphing an equation in 2 variables Graph of an equation in 2 variables is the collection of all points (x,y) whose coordinates are solutions of the equation. How to Graph a function??? 1.Construct a table of values 2.Graph enough solutions to recognize a pattern 3.Connect the points with a line or curve 28

29. Graph the function: f(x) = x + 1 or, y = x + 1 Step1: Table of values of y=x+1 Step2: Plotting the point and Graphing XYOrder pair -3-2(-3,-2) -2(-2,-1) 0(-1,0) 01(0,1) 12(1,2) 23(2,3) 34(3,4) 29

30. Compare graphs with the graph f(x) = x. Graph the function f(x) = x + 3, then compare it to the another function g(x) = x. g(x) = x f(x) = x + 3 The graphs of g(x) and f(x) have the same slope of 1. g(x) = x xf(x) -5 -2 00 11 33 xf(x) -5-2 1 03 14 36 f(x) = x + 3 30

31. Compare graphs with the graph f(x) = x. Graph the function h(x) = 2x, then compare it to the another function f(x) = x. xh(x) -3-6 -2-4 00 24 36 f(x) = x xf(x) -5 -2 00 11 33 h(x) = 2x The graphs of h(x) and f(x) both have a y-int of 0. The slope of h(x) is 2 and therefore is steeper than f(x) with a slope of 1. 31

32. Graphing Quadratic Function y = ax 2 + bx + c 32

33. Graphing Quadratic Functions The graph of a quadratic function is a parabola. A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex. If the parabola opens down, the vertex is the highest point. y x Vertex Standard form of quadratic function is y.= ax 2 +bx+c 33

34. y = ax 2 + bx + c The parabola will open down when the a value is negative. The parabola will open up when the a value is positive. Graphing Quadratic Functions Standard Form a > 0 a < 0 y x 34

35. y x Graphing Quadratic Functions Line of Symmetry Parabolas have a symmetric property to them. If we draw a line down the middle of the parabola, we could fold the parabola in half. The line of symmetry ALWAYS passes through the vertex. Line of Semmetry 35

36. Graphing Quadratic Functions Finding the Line of Symmetry and vertex When a quadratic function is in standard form The equation of the line of symmetry is y = ax 2 + bx + c, (the opposite of b divided by the quantity of 2 times a). We know the line of symmetry always goes through the vertex. Thus, the line of symmetry gives us the x – coordinate of the vertex. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. 36

37. Graphing Quadratic Functions There are 3 steps to graphing a parabola in standard form. STEP 1: Find the line of symmetry STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE 37

38. STEP 1: Find the line of symmetry Let's Graph ONE! y = 2x 2 – 4x – 1 A Quadratic Function Standard Form Thus the line of symmetry is x = 1 38

39. STEP 2: Find the vertex A Quadratic Function in Standard Form Thus the vertex is (1,–3). 39

40. 5 –1 STEP 3: Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. A Quadratic Function in Standard Form 3 2 yx 40

41. Graph of f(x) and f -1 (x) If f is one-to-one function, The graphs of y=f(x) and y=f -1 (x) are reflection of each other in the line y=x 41 Line segment y=x

42. Graph of f(x) and f -1 (x) 42 Figure 1 Figure 2

43. Graph of Function Work on paper: Sketch the graph of the following functions. Sketch the following functions: f:x→ 4x, xєR f:x→ X 2, xєR and x≠0 Find the inverse function f:x→1-3x, xєR and sketch the graph of y=f(x) and y= f -1 (x) 43

44. References / for exercise Cambridge University Press ‘A Level Mathematics, Pure Mathematics 1’ page: 32 and 169 As Level Mathematics 9709- Past papers compilation 44

45. 发展经济学 45 Thank You ! Prakash Adhikari  -Prakash.adhikari@islingtoncollege.edu.np धन्यबाद ! 45