Ex: Restrict the Domain to Make a Function 1 to 1, Then Find the Inverse — Transcript

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1. 0:00- GIVEN F OF X = THE QUANTITY X + 2 SQUARED,
2. 0:04WE WANT TO DETERMINE THE DOMAIN SO F OF X IS INCREASING
3. 0:07AND 1 TO 1.
4. 0:08WE ALSO WANT TO GIVE THE RANGE AND USE INTERVAL NOTATION.
5. 0:13SO HERE'S A GRAPH OF OUR FUNCTION F OF X.
6. 0:16NOTICE IF WE DON'T RESTRICT THE DOMAIN,
7. 0:18THIS FUNCTION IS NOT ONE TO ONE
8. 0:19BECAUSE HORIZONTAL LINES WOULD INTERSECT THIS GRAPH
9. 0:23IN MORE THAN ONE POINT.
10. 0:26NOTICE THE FUNCTION IS ALSO DECREASING ON THE LEFT
11. 0:28AND INCREASING ON THE RIGHT.
12. 0:31SO NOTICE IF WE CONSIDER THIS FUNCTION
13. 0:32ONLY FROM THE VERTEX TO THE RIGHT,
14. 0:35THE FUNCTION IS INCREASING
15. 0:38AND IT'S ALSO 1 TO 1 BECAUSE HORIZONTAL LINES
16. 0:41WOULD ONLY INTERSECT THIS HALF OF THE GRAPH AT ONE POINT.
17. 0:45SO NOW WE'LL DETERMINE THE DOMAIN AND RANGE
18. 0:47IF WE ONLY WANT THIS HALF OF THE GRAPH.
19. 0:50WELL, THE DOMAIN IS A SET OF ALL POSSIBLE X VALUES,
20. 0:53SO IF WE PROJECT THIS GRAPH UNDER THE X AXIS,
21. 0:55NOTICE HOW THE DOMAIN WOULD BE FROM -2 TO THE RIGHT
22. 0:59OR FROM -2 TO INFINITY.
23. 1:02AND IT WOULD INCLUDE THE VERTEX,
24. 1:04SO WE'LL INCLUDE -2 IN THE DOMAIN.
25. 1:07SO THE DOMAIN, USING INTERVAL NOTATION,
26. 1:12WOULD BE FROM -2 TO INFINITY.
27. 1:15AND IT'S CLOSED ON -2 MEANING IT INCLUDES -2.
28. 1:19WE COULD ALSO EXPRESS THIS USING INEQUALITIES
29. 1:22AS X IS GREATER THAN OR = TO -2.
30. 1:26NOW LET'S CONSIDER THE RANGE.
31. 1:28THE RANGE IS A SET OF ALL POSSIBLE Y VALUES
32. 1:30OR OUTPUTS OF THIS FUNCTION ON THE RESTRICTED DOMAIN.
33. 1:35WELL, IF WE PROJECT THIS GRAPH ON TO THE Y AXIS,
34. 1:38NOTICE HOW THE SMALLEST Y VALUE WOULD BE 0,
35. 1:41AND FROM THERE IT INCREASES UPWARD TOWARD POSITIVE INFINITY.
36. 1:46NOW THE RANGE WOULD BE THE INTERVAL FROM 0 TO INFINITY,
37. 1:50CLOSED ON 0 MEANING IT INCLUDES 0,
38. 1:53OR WE COULD SAY Y IS GREATER THAN OR = TO 0.
39. 1:58WITH THIS RESTRICTION, THE FUNCTION F IS NOW ONE TO ONE,
40. 2:02SO WE CAN FIND F INVERSE OF X.
41. 2:04TO DO THIS, LET'S FIRST WRITE THE ORIGINAL FUNCTION
42. 2:07REPLACING F OF X WITH Y,
43. 2:09SO WE'D HAVE Y = THE QUANTITY X + 2 SQUARED.
44. 2:15AND THEN TO FIND THE INVERSE,
45. 2:17WE INTERCHANGE THE X AND Y VARIABLES,
46. 2:19AND THEN SOLVE FOR Y.
47. 2:21SO WE HAVE X = QUANTITY Y + 2 SQUARED.
48. 2:27AND NOW WE'LL SOLVE THIS FOR Y.
49. 2:29THE FIRST STEP WE'LL TAKE THE SQUARE ROOT
50. 2:30OF BOTH SIDES OF THIS EQUATION.
51. 2:32SO WE'D HAVE THE SQUARE ROOT OF X EQUALS
52. 2:36THE SQUARE ROOT OF THE QUANTITY Y + 2 SQUARED.
53. 2:41SO WE HAVE THE SQUARE ROOT OF X EQUALS--
54. 2:45NORMALLY THIS WOULD BE THE OPPOSITE VALUE OF Y + 2,
55. 2:48BUT BECAUSE OF THE RESTRICTIONS HERE
56. 2:50WE DON'T HAVE TO WORRY ABOUT THAT.
57. 2:51THIS WOULD JUST BE Y + 2.
58. 2:54LAST STEP WE'LL SUBTRACT 2 ON BOTH SIDES,
59. 2:56WE HAVE THE SQUARE ROOT OF X - 2 = Y.
60. 3:01THIS IS OUR INVERSE FUNCTION SOLVE FOR Y.
61. 3:04SO WE'LL GO AHEAD AND REPLACE Y WITH F INVERSE OF X.
62. 3:07F INVERSE OF X IS EQUAL TO THE SQUARE ROOT OF X - 2.
63. 3:14WE'RE ALSO ASKED TO GIVE THE DOMAIN AND RANGE.
64. 3:16LET'S GO AHEAD AND DO THAT.
65. 3:17BECAUSE THIS IS THE INVERSE OF FUNCTION F,
66. 3:19THE DOMAIN OF F IS GOING TO BE THE RANGE OF F INVERSE,
67. 3:23AND THE RANGE OF F WILL BE THE DOMAIN OF F INVERSE.
68. 3:27SO THE DOMAIN WILL BE FROM 0 TO INFINITY, CLOSED ON 0,
69. 3:33AND THE RANGE WILL BE THE INTERVAL FROM -2 TO INFINITY.
70. 3:40LET'S GO AHEAD AND FINISH BY VERIFYING THIS GRAPHICALLY.
71. 3:43WE KNOW THAT IF WE GRAPH FUNCTION F
72. 3:45ON THE RESTRICTED DOMAIN
73. 3:47AND WE GRAPH THE INVERSE FUNCTION ON ITS DOMAIN.
74. 3:50THE TWO FUNCTIONS SHOULD BE SYMMETRICAL
75. 3:52ACROSS THE LINE Y = X.
76. 3:56AND HERE'S THE GRAPH OF THE ORIGINAL FUNCTION
77. 3:59ON THE RESTRICTED DOMAIN,
78. 4:03AND HERE'S THE GRAPH OF THE INVERSE FUNCTION
79. 4:06GRAPHED OVER ITS DOMAIN.
80. 4:08NOW WE CAN SEE THAT THESE TWO GRAPHS ARE SYMMETRICAL
81. 4:11ACROSS THE LINE Y = X.
82. 4:19OKAY, I HOPE YOU FOUND THIS HELPFUL.