General Mathematics 11 -review on functions

General Mathematics 11 -review on functions

1. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS

2. What is a function?

3. Definition of a Function: It is a relation define as a set of ordered pairs (x, y) where no two or more distinct ordered pairs have the same first element (x). Every value of x corresponds to a unique value of y

4. Examples: Illustrations below are examples of a function

5. Is it a function or not?

6. What is the difference between a function and a relation?

7. RELATIONS versus FUNCTIONS RELATIONS FUNCTIONS A relation is a rule that relates values from a set of values called the domain to a second set of values called the range. A function is a relation where each element in the domain is related to only one value in the range by some rule.

8. RELATIONS versus FUNCTIONS RELATIONS FUNCTIONS The elements of the domain can be imagines as input to a machine that applies rule to these inputs to generate one or more outputs. The elements of the domain can be imagined as input to a machine that applies a rule so that each input corresponds to only one output.

9. RELATIONS versus FUNCTIONS RELATIONS FUNCTIONS A relation is also a set of ordered pairs (x, y). A function is a set of ordered pairs (x, y) such that no two ordered pairs have the same x-value but different y-values.

10. Is it a function or not? a. f = {(0, -1), (2, -5), (4, -9), (6,-13)} b. r ={(a, 0), (b, -1), (c, 0), (d, -1)} c. g = (5, -10), (25, -50), (50, -100) d. t = {(-2, 0), (-1, 1), (0, 1), (-2, 2)}

11. The function as a machine… We will try to represent mathematical relations as machines with an input and an output, and that the output is related to the input by some rule.

12. Determine if this machine produces a function…

13. Determine if this machine produces a function…

14. Determine if this machine produces a function…

15. Determine if this machine produces a function…

16. What is a table of values?

17. Table of Values a A table of values is commonly observed when describing a function. This shows the correspondence between a set of values of x and a set of values of y in a tabular form.

18. Examples of Table of Values a x 1 4 9 16 y - 5 - 4 - 1 4 11 x -1 -1/4 1/4 1 y -1 - 1/2 1/2 1

19. Is it a function or not? A jeepney and its plate number A student and his ID number A teacher and his cellular phone A pen and the color of its ink

20. What is a vertical line test?

21. Vertical Line Test a The vertical line test for a function states that if each vertical line intersects a graph in the x-y plane at exactly one point, then the graph illustrates a function.

22. Is this a function or not?

23. Is this a function or not?

24. Is this a function or not?

25. Relationship Between the Independent and Dependent Variables

26. Examples: Find the value of y in the equation y = 10x – 3 if x = - 5. Find the value of x if the value of y in the equation is 2.  

27. Applications: A car has travelled a distance of 124 kilometers in 4 hours. Find the speed of the car. The volume of the cube is defined by the function where s is the length of the edge. What is the volume of the cube if the length of the edge is 5 cm? What is the length of its edge if its volume is 728 cubic meters?  

28. REVIEW ON FUNCTIONS Module 1 EVALUATING FUNCTIONS

29. Evaluating Functions a It is the process of determining the value of the function at the number assigned to a given variable.

30. Example: Let . Find the following values of the function f (2) f (-1) f (0) f (- ½ ) f (- 4)  

31. Example: Let . Find the following values of the function g (2) g (4) g (0) g (9) g (- 1/3)  

32. Example: Let h . Find the following values of the function h (1) h (-2) h (6) h (0) h (2)  

33. REVIEW ON FUNCTIONS Module 1 DOMAIN AND RANGE OF FUNCTIONS

34. Domain D of a Function a It is the set of all x-coordinates in the set of ordered pairs. Range R of a Function a It is the set of all y-coordinates in the set of ordered pairs.

35. Determine the domain and the range of the following: a x 1 4 9 16 y - 5 - 4 - 1 4 11 x -1 -1/4 1/4 1 y -1 - 1/2 1/2 1

36. More on Independent Variables a There are instances in which not all values of the independent variables are permissible. That is, some functions have restrictions.

37. Determine the domain and the range of the following: a  

38. Piece-wise Functions a These are functions which are defined in defined in different domains since they are determined by several equations.

39. Determine the domain and the range of the following: a   { 2x + 3 if x ≠ 2 4 if x = 2 { 2x + 3 if x < 1 – if x 1  

40. REVIEW ON FUNCTIONS Module 1 OPERATIONS ON FUNCTIONS

41. Operations on Functions If f and g are functions then (f + g) = f(x) + g(x) (f – g) = f(x)– g(x) (f g) = f(x) g(x) where g(x) ≠ 0  

42. Example Let f(x) = and g(x)= x – 1. Perform the operations and identify the domain (f + g) (f – g) (f g)  

43. Example Let f(x)= x – 3 and g(x) = . Perform the operations and identify the domain (f + g) (f – g) (f g)  

44. REVIEW ON FUNCTIONS Module 1 COMPOSITE FUNCTIONS

45. Operations on Functions If f and g are functions then the composite function denoted by , is defined by  

46. Operations on Functions The domain of is the set of all numbers x in the domain of g such that g(x) is in the domain of f.  

47. Example Let f(x)= x – 3 and g(x) = . Find )(x) )(x) )(3) )(- 4)  

48. REVIEW ON FUNCTIONS Module 1 EVEN AND ODD FUNCTIONS

49. Even and Odd Functions A function f is said to be even if f(–x)=f(x) for each value of x in the domain of f. A function f is said to be odd if f(–x)= – f(x) for each value of x in the domain of f.

50. Example Determine whether each of the following functions is even, odd or neither