WEBVTT 00:00:00.660 --> 00:00:03.697 SALMAN KHAN: I'm here with Jesse Roe of Summit Prep. 00:00:03.697 --> 00:00:04.780 What classes do you teach? 00:00:04.780 --> 00:00:07.040 JESSE ROE: I teach algebra, geometry, and algebra II. 00:00:07.040 --> 00:00:08.270 SALMAN KHAN: And now you're with us, luckily, 00:00:08.270 --> 00:00:10.240 for the summer, doing a whole bunch of stuff 00:00:10.240 --> 00:00:11.115 as a teaching fellow. 00:00:11.115 --> 00:00:12.656 JESSE ROE: Yeah, as a teaching fellow 00:00:12.656 --> 00:00:14.760 I've been helping with organizing and developing 00:00:14.760 --> 00:00:17.294 new content, mostly on the exercise side of the site. 00:00:17.294 --> 00:00:19.710 SALMAN KHAN: And the reason why we're doing this right now 00:00:19.710 --> 00:00:23.020 is you had some very interesting ideas or questions. 00:00:23.020 --> 00:00:25.020 JESSE ROE: Yeah, so as an algebra teacher, 00:00:25.020 --> 00:00:27.710 when I introduce that concept of algebra to students, 00:00:27.710 --> 00:00:28.930 I get a lot of questions. 00:00:28.930 --> 00:00:30.470 One of those questions is, what's 00:00:30.470 --> 00:00:33.280 the difference between an equation and a function? 00:00:33.280 --> 00:00:36.720 SALMAN KHAN: The difference between an equation verses 00:00:36.720 --> 00:00:39.680 a function, that's an interesting question. 00:00:39.680 --> 00:00:42.142 Let's pause it and let the viewers 00:00:42.142 --> 00:00:43.600 try to think about it a little bit. 00:00:43.600 --> 00:00:45.670 And then maybe we'll give a stab at it. 00:00:45.670 --> 00:00:46.670 JESSE ROE: Sounds great. 00:00:49.340 --> 00:00:51.320 So Sal, how would you answer this question? 00:00:51.320 --> 00:00:53.720 What's the difference between an equation and a function? 00:00:53.720 --> 00:00:55.950 SALMAN KHAN: Let me think about it a little bit. 00:00:55.950 --> 00:00:56.660 So let me think. 00:00:56.660 --> 00:00:58.540 I think there's probably equations 00:00:58.540 --> 00:01:00.780 that are not functions and functions that 00:01:00.780 --> 00:01:01.530 are not equations. 00:01:01.530 --> 00:01:03.571 And then there are probably things that are both. 00:01:03.571 --> 00:01:05.330 So let me think of it that way. 00:01:05.330 --> 00:01:08.170 So I'm going to draw-- if this is the world of equations 00:01:08.170 --> 00:01:12.555 right over here, so this is equations. 00:01:15.310 --> 00:01:19.460 And then over here is the world of functions. 00:01:19.460 --> 00:01:21.306 That's the world of functions. 00:01:21.306 --> 00:01:22.680 I do think there is some overlap. 00:01:22.680 --> 00:01:25.520 We'll think it through where the overlap is, 00:01:25.520 --> 00:01:26.717 the world of functions. 00:01:26.717 --> 00:01:29.050 So an equation that is not a function that's sitting out 00:01:29.050 --> 00:01:35.490 here, a simple one would be something like x plus 3 00:01:35.490 --> 00:01:37.760 is equal to 10. 00:01:37.760 --> 00:01:40.354 I'm not explicitly talking about inputs and outputs 00:01:40.354 --> 00:01:41.770 or relationship between variables. 00:01:41.770 --> 00:01:44.282 I'm just stating an equivalence. 00:01:44.282 --> 00:01:48.230 The expression x plus 3 is equal to 10. 00:01:48.230 --> 00:01:50.870 So this, I think, traditionally would just be an equation, 00:01:50.870 --> 00:01:52.480 would not be a function. 00:01:52.480 --> 00:01:55.450 Functions essentially talk about relationships 00:01:55.450 --> 00:01:56.200 between variables. 00:01:56.200 --> 00:01:58.410 You get one or more input variables, 00:01:58.410 --> 00:02:01.590 and we'll give you only one output variable. 00:02:01.590 --> 00:02:04.900 I'll put value. 00:02:04.900 --> 00:02:06.232 And you can define a function. 00:02:06.232 --> 00:02:07.440 And I'll do that in a second. 00:02:07.440 --> 00:02:09.030 You could define a function as an equation, 00:02:09.030 --> 00:02:10.449 but you can define a function a whole bunch of ways. 00:02:10.449 --> 00:02:12.160 You can visually define a function, 00:02:12.160 --> 00:02:16.790 maybe as a graph-- so something like this. 00:02:16.790 --> 00:02:19.050 And maybe I actually mark off the values. 00:02:19.050 --> 00:02:21.320 So that's 1, 2, 3. 00:02:21.320 --> 00:02:23.510 Those are the potential x values. 00:02:23.510 --> 00:02:25.605 And then on the vertical axis, I show 00:02:25.605 --> 00:02:27.480 what the value of my function is going to be, 00:02:27.480 --> 00:02:29.000 literally my function of x. 00:02:29.000 --> 00:02:32.520 And maybe that is 1, 2, 3. 00:02:32.520 --> 00:02:34.200 And maybe this function is defined 00:02:34.200 --> 00:02:36.670 for all non-negative values. 00:02:36.670 --> 00:02:38.690 So this is 0 of x. 00:02:38.690 --> 00:02:42.530 And so let me just draw-- so this right over here, 00:02:42.530 --> 00:02:45.010 at least for what I've drawn so far, defines that function. 00:02:45.010 --> 00:02:46.676 I didn't even have to use an equal sign. 00:02:46.676 --> 00:02:51.732 If x is 2, at least the way I drew it, y is equal to 3. 00:02:51.732 --> 00:02:52.690 You give me that input. 00:02:52.690 --> 00:02:55.877 I gave you the value of only one output. 00:02:55.877 --> 00:02:57.960 So that would be a legitimate function definition. 00:02:57.960 --> 00:02:59.335 Another function definition would 00:02:59.335 --> 00:03:02.050 be very similar to what you do in a computer program, 00:03:02.050 --> 00:03:12.380 something like, let's say, that you input the day of the week. 00:03:12.380 --> 00:03:21.660 And if day is equal to Monday, maybe you output cereal. 00:03:21.660 --> 00:03:24.480 So that's what we're going to eat that day. 00:03:24.480 --> 00:03:30.890 And otherwise, you output meatloaf. 00:03:30.890 --> 00:03:34.590 So this would also be a function. 00:03:34.590 --> 00:03:35.590 We only have one output. 00:03:35.590 --> 00:03:38.420 For any one day of the week, we can only 00:03:38.420 --> 00:03:40.060 tell you cereal or meatloaf. 00:03:40.060 --> 00:03:42.510 There's no days where you are eating both cereal 00:03:42.510 --> 00:03:46.590 and meatloaf, which sounds repulsive. 00:03:46.590 --> 00:03:49.114 And then if I were to think about something 00:03:49.114 --> 00:03:50.780 that could be an equation or a function, 00:03:50.780 --> 00:03:53.300 I guess the way I think about it is an equation is something 00:03:53.300 --> 00:03:55.210 that could be used to define a function. 00:03:55.210 --> 00:04:03.200 So for example, we could say that y is equal to 4x minus 10. 00:04:03.200 --> 00:04:06.840 This is a potential definition for defining y 00:04:06.840 --> 00:04:08.000 as a function of x. 00:04:08.000 --> 00:04:10.400 You give me any value of x. 00:04:10.400 --> 00:04:13.180 Then I can find the corresponding value of y. 00:04:13.180 --> 00:04:16.589 So this is at least how I would think about it.