Upcoming SlideShare
×

# 11 x1 t02 08 inverse functions (2013)

514 views

Published on

Published in: Education
0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total views
514
On SlideShare
0
From Embeds
0
Number of Embeds
235
Actions
Shares
0
18
0
Likes
0
Embeds 0
No embeds

No notes for slide

### 11 x1 t02 08 inverse functions (2013)

1. 1. Inverse Relations
2. 2. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  y
3. 3. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relation
4. 4. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relation
5. 5. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x.
6. 6. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. e.g. y  x 2
7. 7. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. e.g. y  x 2 domain: all real x range: y  0
8. 8. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. e.g. y  x 2 inverse relation: x  y 2 domain: all real x range: y  0
9. 9. Inverse RelationsIf y = f(x) is a relation, then the inverse relation obtained byinterchanging x and y is x = f(y)e.g. y  x 3  x inverse relation is x  y 3  yThe domain of the relation is the range of its inverse relationThe range of the relation is the domain of its inverse relationA relation and its inverse relation are reflections of each other inthe line y = x. e.g. y  x 2 inverse relation: x  y 2 domain: all real x domain: x  0 range: y  0 range: all real y
10. 10. Inverse Functions
11. 11. Inverse FunctionsIf an inverse relation of a function, is a function, then it is calledan inverse function.
12. 12. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test
13. 13. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique.
14. 14. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y x
15. 15. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y x
16. 16. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y xOnly has aninverse relation
17. 17. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y xOnly has an ORinverse relation x  y2 y x
18. 18. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y xOnly has an ORinverse relationx  y2 y x NOT UNIQUE
19. 19. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y ii  y  x 3 y x xOnly has an ORinverse relationx  y2 y x NOT UNIQUE
20. 20. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y ii  y  x 3 y x xOnly has an ORinverse relationx  y2 y x NOT UNIQUE
21. 21. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y ii  y  x 3 y x xOnly has an OR Has aninverse relationx  y2 inverse function y x NOT UNIQUE
22. 22. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y ii  y  x 3 y x xOnly has an OR OR Has aninverse relationx  y2 x  y3 inverse function y x y3 x NOT UNIQUE
23. 23. Inverse Functions If an inverse relation of a function, is a function, then it is called an inverse function.Testing For Inverse Functions(1) Use a horizontal line test OR2 When x  f  y  is rewritten as y  g  x , y  g  x  is unique. i  y  x 2 y ii  y  x 3 y x xOnly has an OR OR Has aninverse relationx  y2 x  y3 inverse function y x y3 x NOT UNIQUE UNIQUE
24. 24. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has aninverse function), then; f 1  f  x   x AND f  f 1  x   x
25. 25. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. x2 f  x  x2
26. 26. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. x2 f  x  x2 x2 y2y x x2 y2
27. 27. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. x2 f  x  x2 x2 y2y x x2 y2  y  2 x  y  2 xy  2 x  y  2  x  1 y  2 x  2 2x  2 y 1 x
28. 28. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. x2  x22 f  x  2  x2 x 2 f 1  f  x     x2 x2 y2 1   y x  x 2 x2 y2  y  2 x  y  2 xy  2 x  y  2  x  1 y  2 x  2 2x  2 y 1 x
29. 29. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. x2  x22 f  x  2  x2 x 2 f 1  f  x     x2 x2 y2 1  y x  x 2 x2 y2 2x  4  2x  4  y  2 x  y  2  x2 x2 xy  2 x  y  2 4x  x  1 y  2 x  2  4 2x  2 x y 1 x
30. 30. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. x2  x22  2x  2   2 f  x  2    x2 x 2 1 x  f 1  f  x     f  f 1  x     x2  2x  2   2 x2 y2 1    y x  x 2  1 x  x2 y2 2x  4  2x  4  y  2 x  y  2  x2 x2 xy  2 x  y  2 4x  x  1 y  2 x  2  4 2x  2 x y 1 x
31. 31. If the inverse relation of y= f(x) is a function, (i.e. y = f(x) has an inverse function), then; f 1  f  x   x AND f  f 1  x   xe.g. x2  x22  2x  2   2 f  x  2    x2 x 2 1 x  f 1  f  x     f  f 1  x     x2  2x  2   2 x2 y2 1    y x  x 2  1 x  x2 y2 2x  4  2x  4 2x  2  2  2x  y  2 x  y  2   x2 x2 2x  2  2  2x xy  2 x  y  2 4x 4x  x  1 y  2 x  2   4 4 2x  2 x x y 1 x
32. 32. (ii) Draw the inverse relation y x
33. 33. (ii) Draw the inverse relation y x
34. 34. (ii) Draw the inverse relation y x
35. 35. (ii) Draw the inverse relation y x Exercise 2H; 1aceg, 2, 3bdf, 5ac, 6bd, 7ac, 9bde, 10adfhj