Grade 12 Functions and Inverses This is a build-up from the foundation laid in earlier grades. The concept of the function introduced in grade 10 forms an important foundation for this topic.

1. FUNCTIONS
GRADE 12
VUKILE XHEGO (GDE)

2. REVISION AND SUMMARY OF GRADE 11 FUNCTIONS
VUKILE XHEGO (GDE)

3. REVISION AND SUMMARY OF GRADE 11 FUNCTIONS
VUKILE XHEGO (GDE)

4. REVISION AND SUMMARY OF GRADE 11 FUNCTIONS
VUKILE XHEGO (GDE)

5. INVERSE
FUNCTIONS
GRADE 12
VUKILE XHEGO (GDE)

6. REVISION OF THE CONCEPT OF A FUNCTION
A function is a rule by means of which each element of the domain is
associated with only one element of the range.
For a function, two or more elements of the domain may be associated
with the same element of the range as well.
However, a relation is not a function if one element of the domain is
associated with more than one element of the range.
VUKILE XHEGO (GDE)

7. REVISION OF THE CONCEPT OF A FUNCTION
Consider the following relations.
VUKILE XHEGO (GDE)

8. REVISION OF THE CONCEPT OF A FUNCTION
VUKILE XHEGO (GDE)

9. REVISION OF THE CONCEPT OF A FUNCTION
VUKILE XHEGO (GDE)

10. The Vertical and Horizontal Line Tests
We can use a ruler to perform the “vertical line test” on a graph to see
whether it is a function or not.
Hold a clear plastic ruler parallel to the y-axis, i.e. vertical.
Move it from left to right over the axes.
If the ruler only ever cuts the curve at one place as the ruler moves
from left to right across the graph, then the graph is a function.
If the ruler at any stage cuts the graph at more than one place, then the
graph is not a function. This is because the same x-value will be
associated with more than one y-value.
VUKILE XHEGO (GDE)

11. The Vertical and Horizontal Line Tests
Once a graph passes the “vertical line test”, the “horizontal line test” is
used to determine the type of function.
We have two types of functions:
1. One-to-one function
2. Many-to-one function
VUKILE XHEGO (GDE)

12. The Vertical and Horizontal Line Tests
Procedure of “horizontal line test”:
Place the ruler horizontally so that it is parallel to the x-axis
Move the ruler up and down
If the ruler only cuts the curve in one place as the ruler moves up and
down the graph, then the graph is a one-to-one function.
If the ruler at any stage cuts the graph in more than one place, then the
graph is a many-to-one function.
VUKILE XHEGO (GDE)

13. The Vertical and Horizontal Line Tests
VUKILE XHEGO (GDE)

14. The Vertical and Horizontal Line Tests
VUKILE XHEGO (GDE)

15. The Vertical and Horizontal Line Tests
VUKILE XHEGO (GDE)

16. The Vertical and Horizontal Line Tests
VUKILE XHEGO (GDE)

17. VUKILE XHEGO (GDE)

18. VUKILE XHEGO (GDE)

19. INVERSE FUNCTIONS
INVERSES OF ONE-TO-ONE LINEAR FUNCTIONS
VUKILE XHEGO (GDE)

20. Mapping and functional notation
VUKILE XHEGO (GDE)

21. Mapping and functional notation
VUKILE XHEGO (GDE)

22. Inverses of One-to-One Linear Functions
VUKILE XHEGO (GDE)

23. Inverses of One-to-One Linear Functions
VUKILE XHEGO (GDE)

24. Inverses of One-to-One Linear Functions
VUKILE XHEGO (GDE)

25. Inverses of One-to-One Linear Functions
VUKILE XHEGO (GDE)

26. Inverses of One-to-One Linear Functions
VUKILE XHEGO (GDE)

27. VUKILE XHEGO (GDE)

28. INVERSES OF MANY-TO-ONE QUADRATIC FUNCTIONS
VUKILE XHEGO (GDE)

29. INVERSES OF MANY-TO-ONE QUADRATIC FUNCTIONS
VUKILE XHEGO (GDE)

30. INVERSES OF MANY-TO-ONE QUADRATIC FUNCTIONS
VUKILE XHEGO (GDE)

31. Example 5: Solutions
VUKILE XHEGO (GDE)

32. INVERSES OF MANY-TO-ONE QUADRATIC FUNCTIONS
VUKILE XHEGO (GDE)

33. INVERSES OF MANY-TO-ONE QUADRATIC FUNCTIONS
VUKILE XHEGO (GDE)

34. Restricting the domain and range
VUKILE XHEGO (GDE)

35. Example 1: Solutions
VUKILE XHEGO (GDE)

36. Example 1: Solutions
VUKILE XHEGO (GDE)

37. Restricting the Domain and Range
VUKILE XHEGO (GDE)

38. Example 2: Solutions
VUKILE XHEGO (GDE)

39. VUKILE XHEGO (GDE)

40. THE INVERSE OF THE EXPONENTIAL FUNCTION
Introduction
VUKILE XHEGO (GDE)

41. THE INVERSE OF THE EXPONENTIAL FUNCTION
Finding the equation of the Inverse
Step 1: let f(x)=y
𝑦 = 2𝑥
Step 2: interchange (swap) x and y
𝑥 = 2𝑦
We have a problem…
None of the methods learned so far will help make y the subject
VUKILE XHEGO (GDE)

42. THE INVERSE OF THE
EXPONENTIAL
FUNCTION
A Scottish mathematician named John
Napier (1550-1617) devised a clever way
of making y the subject of the formula.
He introduced a notation referred to as a
logarithm.
We will now discuss the concept of a
logarithm and then later on develop the
theory of logarithms in more detail.
VUKILE XHEGO (GDE)

43. Introduction to Logarithms
If a number is written in exponential form, then the exponent is called the
logarithm of the number.
For example, the number 8 can be written in exponential form as 8 = 23
.
Clearly, the exponent in this example is 3 and the base is 2.
We can then say that the logarithm of 8 to base 2 is 3.
This can be written as log2 8 = 3. The base 2 is written as a sub-script
between the “log” and the number 8.
VUKILE XHEGO (GDE)

44. Introduction to Logarithms
Examples
VUKILE XHEGO (GDE)

45. Introduction to Logarithms
Examples
VUKILE XHEGO (GDE)

46. VUKILE XHEGO (GDE)

47. LAWS OF LOGARITHMS
VUKILE XHEGO (GDE)

48. LAWS OF LOGARITHMS
VUKILE XHEGO (GDE)

49. LAWS OF LOGARITHMS
VUKILE XHEGO (GDE)

50. VUKILE XHEGO (GDE)

51. The
Logarithmic
Function
VUKILE XHEGO (GDE)

52. THE INVERSE OF THE EXPONENTIAL FUNCTION
Introduction
VUKILE XHEGO (GDE)

53. THE INVERSE OF THE EXPONENTIAL FUNCTION
VUKILE XHEGO (GDE)

54. The Logarithmic Function
VUKILE XHEGO (GDE)

55. The Logarithmic Function
VUKILE XHEGO (GDE)

56. The Logarithmic Function
Summary of the two types of logarithmic functions:
VUKILE XHEGO (GDE)

57. Example 1
VUKILE XHEGO (GDE)

58. Example 1: solutions
VUKILE XHEGO (GDE)

59. Example 1 (solutions)
VUKILE XHEGO (GDE)

60. Example 2
VUKILE XHEGO (GDE)

61. Example 3
VUKILE XHEGO (GDE)

62. Example 3
VUKILE XHEGO (GDE)

63. Example 4
VUKILE XHEGO (GDE)

64. Example 4
VUKILE XHEGO (GDE)

65. Example 5
VUKILE XHEGO (GDE)

66. Example 5: solutions
VUKILE XHEGO (GDE)

67. VUKILE XHEGO (GDE)

68. VUKILE XHEGO (GDE)

69. VUKILE XHEGO (GDE)