Lesson: Inverse Functions — Transcript

Full transcript

1. 0:06- HELLO AND WELCOME TO A VIDEO ON INVERSE FUNCTIONS.
2. 0:08THE GOALS OF THE VIDEO ARE TO DEFINE INVERSE FUNCTIONS,
3. 0:11DETERMINE IF A FUNCTION HAS AN INVERSE FUNCTION,
4. 0:15AND LASTLY, TO DETERMINE THE EQUATION OF AN INVERSE FUNCTION.
5. 0:20SO WE'RE GOING TO START OFF
6. 0:21BY TALKING ABOUT A SPECIAL TYPE OF FUNCTION,
7. 0:23A ONE-TO-ONE FUNCTION.
8. 0:25IF A FUNCTION IS DEFINED
9. 0:27SO THAT EACH RANGE ELEMENT OR Y VALUE IS USED ONLY ONCE
10. 0:32THEN IT IS A ONE-TO-ONE FUNCTION.
11. 0:34IF EVERY Y VALUE IS ONLY PAIRED WITH ONE X VALUE
12. 0:38WE HAVE A SPECIAL TYPE OF FUNCTION
13. 0:40CALLED A ONE-TO-ONE FUNCTION.
14. 0:42AND A HORIZONTAL LINE TEST HELPS DETERMINE GRAPHICALLY
15. 0:45WHETHER A FUNCTION IS ONE-TO-ONE.
16. 0:48IF A HORIZONTAL LINE DOES NOT INTERSECT
17. 0:50THE GRAPH OF A FUNCTION IN MORE THAN ONE POINT
18. 0:54IT IS A ONE-TO-ONE FUNCTION.
19. 0:56AND WHAT'S SO IMPORTANT ABOUT THIS IS THAT
20. 0:57ONLY ONE-TO-ONE FUNCTIONS HAVE INVERSE FUNCTIONS.
21. 1:02SO FOR EXAMPLE ON THIS FIRST GRAPH,
22. 1:04IT PASSES THE VERTICAL LINE TEST SO IT'S A FUNCTION
23. 1:07HOWEVER, IT FAILS THE HORIZONTAL LINE TEST
24. 1:10BECAUSE THESE HORIZONTAL LINES INTERSECT THE GRAPH
25. 1:14IN MORE THAN ONE POINT.
26. 1:15SO THIS FUNCTION IS NOT ONE-TO-ONE.
27. 1:20NOW THE SECOND GRAPH IS A LITTLE BIT OF A TRICK QUESTION
28. 1:22BECAUSE IT'S NOT EVEN A FUNCTION
29. 1:24BECAUSE IT FAILS THE VERTICAL LINE TEST.
30. 1:26SO IF IT'S NOT A FUNCTION
31. 1:28OF COURSE IT CAN'T BE A ONE-TO-ONE FUNCTION.
32. 1:32SO THIS LAST GRAPH IS THE ONLY FUNCTION
33. 1:36THAT IS ALSO A ONE-TO-ONE FUNCTION
34. 1:38BECAUSE THESE HORIZONTAL LINES NEVER INTERSECT THE GRAPH
35. 1:43IN MORE THAN ONE POINT.
36. 1:45SO THIS IS A ONE-TO-ONE FUNCTION.
37. 1:48WHICH MEANS THIS FUNCTION DOES HAVE AN INVERSE FUNCTION.
38. 1:53WELL, NOW LET'S TALK ABOUT WHAT AN INVERSE FUNCTION IS.
39. 1:56INFORMALLY A FUNCTION AND ITS INVERSE UNDO EACH OTHER.
40. 1:59FOR EXAMPLE, F OF X = 3 x X AND G OF X = X DIVIDED BY 3,
41. 2:05WELL MULTIPLYING BY 3 AND DIVIDING BY 3
42. 2:08ARE OPPOSITE OPERATIONS AND THEREFORE UNDO EACH OTHER.
43. 2:12THEREFORE, F OF X AND G OF X ARE INVERSES OF ONE ANOTHER.
44. 2:16YOU CAN ALMOST THINK OF THIS AS A CONVEYOR BELT
45. 2:19WHERE WE PUT AN INPUT HERE
46. 2:22AND WHEN THIS INPUT GOES INTO F
47. 2:23THE OUTPUT WOULD BE 3 x X OR 3 x 2,
48. 2:27THE OUTPUT WOULD BE A 6.
49. 2:29NOW IF THIS OUTPUT BECOMES AN INPUT INTO G
50. 2:32AND G TAKES THE INPUT AND DIVIDES BY 3, 6 DIVIDED BY 3
51. 2:37THE OUTPUT WOULD BE 2.
52. 2:39SO THESE TWO FUNCTIONS UNDO EACH OTHER
53. 2:41BECAUSE WHAT WE START WITH
54. 2:44IS THE SAME THING THAT WE END UP WITH.
55. 2:47ANOTHER EXAMPLE WOULD BE F OF X = X + 3 AND G OF X = X - 3.
56. 2:53OBVIOUSLY + 3 AND - 3 ARE OPPOSITE OPERATIONS
57. 2:56THEREFORE THESE TWO FUNCTIONS UNDO EACH OTHER
58. 2:59AND THEREFORE THEY'RE INVERSES OF ONE ANOTHER.
59. 3:01LET'S TAKE A LOOK AT A MORE FORMAL DEFINITION.
60. 3:04IF F IS A FUNCTION FROM A SET "A" TO A SET B,
61. 3:09THEN AN INVERSE FUNCTION OF F IS A FUNCTION FROM B TO "A",
62. 3:13WITH THE PROPERTY THAT AROUND TRIP OR COMPOSITION
63. 3:17FROM "A" TO B TO "A"
64. 3:19RETURNS EACH ELEMENT OF THE INITIAL SET TO IT'S SELF.
65. 3:23WHICH LEADS US TO F OF G OF X WILL EQUAL G OF F OF X
66. 3:28WHICH EQUALS X.
67. 3:29AND LASTLY, A FUNCTION THAT HAS AN INVERSE
68. 3:31IS CALLED INVERTIBLE
69. 3:33AND THE INVERSE FUNCTION IS THEN UNIQUELY DETERMINED BY F
70. 3:37AND IT'S DENOTED BY THIS INVERSE FUNCTION NOTATION,
71. 3:42AND BE CAREFUL THIS LOOKS VERY SIMILAR TO EXPONENTIAL NOTATION
72. 3:46BUT THIS IS INVERSE NOTATION FOR FUNCTION F.
73. 3:49OKAY, SO NEXT LET'S TALK ABOUT
74. 3:50HOW WE DETERMINE AN INVERSE FUNCTION.
75. 3:53FIRST, WE HAVE TO DETERMINE IF THE FUNCTION IS ONE-TO-ONE.
76. 3:56IF IT'S NOT ONE-TO-ONE IT WILL NOT HAVE AN INVERSE FUNCTION.
77. 4:01STEP TWO, WE'LL INTERCHANGE THE X AND Y VARIABLES
78. 4:04AND THIS NEW FUNCTION IS THE INVERSE FUNCTION.
79. 4:08IF THE RESULT IS AN EQUATION,
80. 4:09WE NEED TO SOLVE THE EQUATION FOR Y
81. 4:12AND THEN REPLACE Y WITH OUR INVERSE FUNCTION NOTATION
82. 4:16SHOWING IT'S THE INVERSE OF F.
83. 4:19NOW OF COURSE WE COULD PERFORM THIS PROCEDURE ON ANY FUNCTION
84. 4:23BUT THE RESULTING INVERSE WILL ONLY BE ANOTHER FUNCTION
85. 4:26IF THE ORIGINAL FUNCTION IS ONE-TO-ONE.
86. 4:30TO EMPHASIS THAT LET'S TAKE A LOOK AT THESE FOUR FUNCTIONS.
87. 4:33WHICH OF THESE ARE ONE-TO-ONE FUNCTIONS
88. 4:36AND WHAT DOES IT TELL US ABOUT THE FUNCTION.
89. 4:38REMEMBER TO DETERMINE IF A FUNCTION IS ONE-TO-ONE,
90. 4:40WE HAVE TO PERFORM THE HORIZONTAL LINE TEST.
91. 4:44SO THIS FUNCTION QUICKLY FAILS THE HORIZONTAL LINE TEST
92. 4:47SO IT'S NOT ONE-TO-ONE.
93. 4:49THIS ONE OBVIOUSLY FAILS AS WELL
94. 4:52AND THESE ARE ON THE RIGHT THEY PASS.
95. 4:54NEVER WILL A HORIZONTAL LINE INTERSECT THE GRAPH
96. 4:57IN MORE THAN ONE POINT.
97. 4:59SO WHAT THAT MEANS IS THESE TWO ON THE RIGHT
98. 5:01ARE ONE-TO-ONE FUNCTIONS
99. 5:03AND THEREFORE THESE TWO HAVE INVERSE FUNCTIONS
100. 5:06AND THESE TWO DO NOT HAVE INVERSE FUNCTIONS.
101. 5:11SO LET'S GO AHEAD AND TRY TO FIND SOME INVERSE FUNCTIONS.
102. 5:15HERE'S A FUNCTION THAT CONSISTS OF THREE ORDERED PAIRS.
103. 5:20SO THE INVERSE FUNCTION WOULD JUST CONSIST
104. 5:22OF THE ORDERED PAIRS (2, 1), (3, -2), AND (-2, 5).
105. 5:31NOTICE HOW WE JUST INTERCHANGED THE X AND THE Y COORDINATES
106. 5:34FOR THESE ORDERED PAIRS
107. 5:36AND WE HAVE THE INVERSE FUNCTION.
108. 5:38NOW WE SHOULD MAKE A NOTE
109. 5:39THAT THEY TOLD US THE ORIGINAL FUNCTIONS WERE ONE-TO-ONE,
110. 5:42THEREFORE WE CAN GO AHEAD AND FIND THE INVERSE FUNCTION
111. 5:45WITHOUT QUESTIONING
112. 5:46WHETHER THEY'RE ONE-TO-ONE TO BEGIN WITH.
113. 5:48NEXT, WE HAVE AN EQUATION THAT WE WANT TO FIND THE INVERSE OF,
114. 5:52SO WE'RE GOING TO INTERCHANGE THE X AND THE Y VARIABLES.
115. 5:54SO WE'LL HAVE X = Y TO THE 3rd + 2.
116. 5:58THIS IS OUR INVERSE FUNCTION
117. 6:01BUT NOW WE DO HAVE TO SOLVE THIS FOR Y.
118. 6:03SO WE'LL SUBTRACT 2 ON BOTH SIDES.
119. 6:06NEXT, TO FIND Y IF WE HAVE Y CUBED
120. 6:10WE'LL TAKE THE CUBE ROOT OF BOTH SIDES
121. 6:11SO WE'LL HAVE Y = THE CUBE ROOT OF X - 2.
122. 6:17NOW THE LAST STEP IS TO REPLACE THIS Y
123. 6:19WITH OUR INVERSE FUNCTION NOTATION.
124. 6:22SO WE'LL HAVE F INVERSE OF X IS EQUAL TO THE CUBE ROOT OF X - 2,
125. 6:29AND TO BE CONSISTENT
126. 6:30LET'S GO AHEAD AND REWRITE OUR ORIGINAL EQUATION
127. 6:31IN FUNCTION NOTATION
128. 6:33AND THIS WOULD HAVE BEEN F OF X = X CUBED + 2.
129. 6:39AND LET'S TAKE A MOMENT
130. 6:40AND LOOK AT THE GRAPHS OF THESE TWO FUNCTIONS TOGETHER.
131. 6:44LET'S GO AHEAD AND TYPE THE ORIGINAL FUNCTION INTO Y1
132. 6:49AND WE'LL TYPE IN OUR INVERSE FUNCTION INTO Y2
133. 6:52SO WE'LL RAISE THE QUANTITY X - 2 TO THE 1/3 POWER.
134. 6:59AND LET'S GO AHEAD AND GRAPH Y = X AS WELL
135. 7:04AND LET'S GO AHEAD AND MAKE SURE WE HAVE THE STANDARD WINDOW,
136. 7:06ZOOM 6.
137. 7:12THERE'S OUR FUNCTION,
138. 7:15THERE'S OUR INVERSE FUNCTION,
139. 7:17AND THESE TWO SHARE A SPECIAL GEOMETRIC PROPERTY
140. 7:20IN RELATION TO THE LINE Y = X.
141. 7:23IT WILL BECOME EVEN MORE OBVIOUS IF WE PRESS ZOOM SQUARE, ZOOM 5
142. 7:31AND SO ANOTHER PROPERTY OF A FUNCTION AND IT'S INVERSE IS
143. 7:34THEY'RE SYMMETRICAL ACROSS THE LINE Y = X.
144. 7:38AND YOU CAN SEE IF WE WERE TO FOLD THESE FUNCTIONS
145. 7:40ACROSS THIS LINE Y = X
146. 7:43THEY'LL MATCH UP PERFECTLY WITH THE OTHER PIECE
147. 7:45ON THE OTHER SIDE OF Y = X.
148. 7:49LET'S TAKE A LOOK AT TWO MORE.
149. 7:51AGAIN, WE HAVE ANOTHER FUNCTION THAT'S GIVEN AS ONE-TO-ONE
150. 7:56SO WE'LL GO AHEAD AND INTERCHANGE OUR VARIABLES.
151. 7:59WELL HAVE X = 2 DIVIDED BY THE QUANTITY Y - 4.
152. 8:04REMEMBER THAT THIS GIVEN EQUATION IS A FUNCTION
153. 8:07WHICH COULD HAVE BEEN WRITTEN AS F OF X = 2 DIVIDED BY X - 4.
154. 8:13WE NEED TO SOLVE THIS FOR Y
155. 8:15SO LET'S GO AHEAD AND PERFORM CROSS PRODUCTS HERE.
156. 8:18X x Y - 4 MUST EQUAL 2.
157. 8:23NOW WE'LL DIVIDE BY X ON BOTH SIDES.
158. 8:27SO WE HAVE Y - 4 = 2 DIVIDED BY X,
159. 8:30ADD 4 TO BOTH SIDES,
160. 8:33AND THE LAST STEP SHOULD BE TO REPLACE OUR Y VARIABLE
161. 8:36WITH INVERSE FUNCTION NOTATION.
162. 8:39SO WE'LL HAVE F INVERSE OF X = 2 DIVIDED BY X + 4.
163. 8:45SO HERE'S OUR INVERSE FUNCTION
164. 8:47AND HERE'S OUR ORIGINAL FUNCTION.
165. 8:49LET'S GO AHEAD AND SUMMARIZE,
166. 8:50IN A ONE-TO-ONE FUNCTION
167. 8:51EACH Y VALUE CORRESPONDS TO ONLY ONE X VALUE
168. 8:56AND IT WOULD ALSO PASS THE VERTICAL
169. 8:57AND HORIZONTAL LINE TEST.
170. 8:59IF A FUNCTION F IS A ONE-TO-ONE
171. 9:02THEN IT DOES HAVE AN INVERSE FUNCTION.
172. 9:04THE DOMAIN OF F IS THE RANGE OF F INVERSE
173. 9:07AND THE RANGE OF F IS THE DOMAIN OF F INVERSE,
174. 9:11AGAIN BECAUSE WE'RE INTERCHANGING
175. 9:12THE X AND THE Y VARIABLES.
176. 9:14AND LASTLY, THE GRAPHS OF F AND F INVERSE
177. 9:18ARE REFLECTIONS OF EACH OTHER ACROSS THE LINE Y = X.
178. 9:23I HOPE YOU FOUND THIS VIDEO HELPFUL,
179. 9:25THANK YOU FOR WATCHING.