FUNCTIONS- GENERAL MATHEMATICS (SENIOR HIGH)

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1. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela

2. GENERAL MATHEMATICS Samar College Galina V. Panela 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 GENERAL MATHEMATICS Samar College Galina V. Panela

4. Examples: Illustrations below are examples of a function GENERAL MATHEMATICS Samar College Galina V. Panela

5. Is it a function or not? GENERAL MATHEMATICS Samar College Galina V. Panela

6. GENERAL MATHEMATICS Samar College Galina V. Panela What is the difference between a function and a relation?

7. RELATIONS versus FUNCTIONS GENERAL MATHEMATICS Samar College Galina V. Panela 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 GENERAL MATHEMATICS Samar College Galina V. Panela 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 GENERAL MATHEMATICS Samar College Galina V. Panela 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)} GENERAL MATHEMATICS Samar College Galina V. Panela

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. GENERAL MATHEMATICS Samar College Galina V. Panela

12. Determine if this machine produces a function… GENERAL MATHEMATICS Samar College Galina V. Panela

13. Determine if this machine produces a function… GENERAL MATHEMATICS Samar College Galina V. Panela

14. Determine if this machine produces a function… GENERAL MATHEMATICS Samar College Galina V. Panela

15. Determine if this machine produces a function… GENERAL MATHEMATICS Samar College Galina V. Panela

16. GENERAL MATHEMATICS Samar College Galina V. Panela 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. GENERAL MATHEMATICS Samar College Galina V. Panela

18. Examples of Table of Values a GENERAL MATHEMATICS Samar College Galina V. Panela 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 GENERAL MATHEMATICS Samar College Galina V. Panela

20. GENERAL MATHEMATICS Samar College Galina V. Panela 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. GENERAL MATHEMATICS Samar College Galina V. Panela

22. Is this a function or not? GENERAL MATHEMATICS Samar College Galina V. Panela

23. Is this a function or not? GENERAL MATHEMATICS Samar College Galina V. Panela

24. Is this a function or not? GENERAL MATHEMATICS Samar College Galina V. Panela

25. Relationship Between the Independent and Dependent Variables GENERAL MATHEMATICS Samar College Galina V. Panela

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.   GENERAL MATHEMATICS Samar College Galina V. Panela

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?   GENERAL MATHEMATICS Samar College Galina V. Panela

28. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela 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. GENERAL MATHEMATICS Samar College Galina V. Panela

30. Example: Let . Find the following values of the function f (2) f (-1) f (0) f (- ½ ) f (- 4)   GENERAL MATHEMATICS Samar College Galina V. Panela

31. Example: Let . Find the following values of the function g (2) g (4) g (0) g (9) g (- 1/3)   GENERAL MATHEMATICS Samar College Galina V. Panela

32. Example: Let h . Find the following values of the function h (1) h (-2) h (6) h (0) h (2)   GENERAL MATHEMATICS Samar College Galina V. Panela

33. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela 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. GENERAL MATHEMATICS Samar College Galina V. Panela

35. Determine the domain and the range of the following: a GENERAL MATHEMATICS Samar College Galina V. Panela 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. GENERAL MATHEMATICS Samar College Galina V. Panela

37. Determine the domain and the range of the following: a   GENERAL MATHEMATICS Samar College Galina V. Panela

38. Piece-wise Functions a These are functions which are defined in defined in different domains since they are determined by several equations. GENERAL MATHEMATICS Samar College Galina V. Panela

39. Determine the domain and the range of the following: a   GENERAL MATHEMATICS Samar College Galina V. Panela { 2x + 3 if x ≠ 2 4 if x = 2 { 2x + 3 if x < 1 – if x 1  

40. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela 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   GENERAL MATHEMATICS Samar College Galina V. Panela

42. Example Let f(x) = and g(x)= x – 1. Perform the operations and identify the domain (f + g) (f – g) (f g )   GENERAL MATHEMATICS Samar College Galina V. Panela

43. Example Let f(x)= x – 3 and g(x) = . Perform the operations and identify the domain (f + g) (f – g) (f g )   GENERAL MATHEMATICS Samar College Galina V. Panela

44. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela COMPOSITE FUNCTIONS

45. Operations on Functions If f and g are functions then the composite function denoted by , is defined by   GENERAL MATHEMATICS Samar College Galina V. Panela

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.   GENERAL MATHEMATICS Samar College Galina V. Panela

47. Example Let f(x)= x – 3 and g(x) = . Find )(x) )(x) )(3) )(- 4)   GENERAL MATHEMATICS Samar College Galina V. Panela

48. REVIEW ON FUNCTIONS Module 1 GENERAL MATHEMATICS Samar College Galina V. Panela 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 . GENERAL MATHEMATICS Samar College Galina V. Panela

50. Example Determine whether each of the following functions is even, odd or neither   GENERAL MATHEMATICS Samar College Galina V. Panela