FUNCTIONS and INVERSES.pdf
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The document provides a comprehensive overview of functions and inverses, specifically targeting grade 11 and 12 students. It covers the definition of functions, including the vertical and horizontal line tests to distinguish between different types of functions. Additionally, it introduces the logarithmic function as the inverse of the exponential function and outlines key concepts and examples related to both functions and their inverses.
FUNCTIONS and INVERSES.pdf
- 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)
- 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)
- 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)
- 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)
- 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)
- 47. LAWS OF LOGARITHMS VUKILE XHEGO (GDE)
- 48. LAWS OF LOGARITHMS VUKILE XHEGO (GDE)
- 49. LAWS OF LOGARITHMS 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)