Recognizing features of functions (example 2) | Algebra II | Khan Academy — Transcript

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1. 0:00Which of these functions is odd?
2. 0:03And so let's remind ourselves what
3. 0:05it means for a function to be odd.
4. 0:07So I have a function-- well, they've
5. 0:09already used f, g, and h, so I'll use j.
6. 0:12So function j is odd.
7. 0:14If you evaluate j at some value-- so let's say j of a.
8. 0:19And if you evaluate that j at the negative of that value,
9. 0:23and if these two things are the negative of each other,
10. 0:27then my function is odd.
11. 0:29If these two things were the same--
12. 0:31if they didn't have this negative here-- then
13. 0:32it would be an even function.
14. 0:34So let's see which of these meet the criteria of being odd.
15. 0:38So let's look at f of x.
16. 0:41So we could pick a particular point.
17. 0:43So let's say when x is equal to 2.
18. 0:44So we get f of 2 is equal to 2.
19. 0:49Now, what is f of negative 2?
20. 0:51f of negative 2 looks like it is 6.
21. 0:55f of negative 2 is equal to 6.
22. 0:58So these aren't the negative of each other.
23. 1:00In order for this to be odd, f of negative 2
24. 1:03would have had to be equal to the negative of this, would
25. 1:05have had to be equal to negative 2.
26. 1:08So f of x is definitely not odd.
27. 1:13So all I have to do is find even one case that
28. 1:16violated this constraint to be odd.
29. 1:18And so I can say it's definitely not odd.
30. 1:22Now let's look at g of x.
31. 1:26So I could use the same-- let's see, when x is equal to 2,
32. 1:30we get g of 2 is equal to negative 7.
33. 1:34Now let's look at when g is negative 2.
34. 1:37So we get g of negative 2 is also equal to negative 7.
35. 1:43So here we have a situation-- and it
36. 1:45looks like that's the case for any x we pick-- that g of x
37. 1:49is going to be equal to g of negative x.
38. 1:51So g of x is equal to g of negative x.
39. 1:56It's symmetric around the y-- or I
40. 1:59should say the vertical axis-- right over here.
41. 2:01So g of x is even, not odd.
42. 2:04So which of these functions is odd?
43. 2:05Definitely not g of x.
44. 2:07So our last hope is h of x.
45. 2:09Let's see if h of x seems to meet the criteria.
46. 2:14I'll do it in this green color.
47. 2:16So if we take h of 1-- and we can look at it even visually.
48. 2:21So h of 1 gets us right over here. h of negative 1
49. 2:25seems to get us an equal amount, an equal distance, negative.
50. 2:30So it seems to fit for 1.
51. 2:32For 2-- well, 2 is at the x-axis.
52. 2:35But that's definitely h of 2 is 0. h of negative 2 is 0.
53. 2:39But those are the negatives of each other.
54. 2:400 is equal to negative 0.
55. 2:42If we go to, say, h of 4, h of 4 is this negative number.
56. 2:48And h of negative 4 seems to be a positive number
57. 2:53of the same magnitude.
58. 2:55So once again, this is the negative of this.
59. 2:59So it looks like this is indeed an odd function.
60. 3:03And another way to visually spot an odd function
61. 3:07is a function-- it's going to go through the origin,
62. 3:10and you could essentially flip it over on both axes.
63. 3:14So if you flip this, the right half, over the left half,
64. 3:17and then flip that over the horizontal axis,
65. 3:20you are going to get this right over here.
66. 3:22So you see here we're going up and to the right.
67. 3:24Here we're going to go down and to the left.
68. 3:26And then you curve right over there.
69. 3:28You curve up just like that.
70. 3:29But the easiest way to test it is just to do what we did,
71. 3:32look at a given x.
72. 3:33So for example, when x is equal to 8, h of 8
73. 3:37looks like this number right around 8.
74. 3:40h of negative 8 looks like it's pretty close to negative 8.
75. 3:44So they seem to be the negative of each other.
76. 3:46It sounds like a car crash just happened outside.
77. 3:48Anyway, hopefully you enjoyed that.
78. 3:50Not the car crash, the math problem.