0:00:00.000,0:00:02.460 In this video, I want[br]to do a few examples 0:00:02.460,0:00:03.800 dealing with functions. 0:00:03.800,0:00:06.570 Functions tend to be something[br]that a lot of students find 0:00:06.570,0:00:09.230 difficult, but I think if you[br]really get what we're talking 0:00:09.230,0:00:11.070 about, you'll see that it's[br]actually a pretty 0:00:11.070,0:00:12.240 straightforward idea. 0:00:12.240,0:00:13.710 And you sometimes wonder,[br]well what was all 0:00:13.710,0:00:14.880 of the hubbub about? 0:00:14.880,0:00:16.720 All a function is, is an 0:00:16.720,0:00:19.830 association between two variables. 0:00:19.830,0:00:25.540 So if we say that y is equal to[br]a function of x, all that 0:00:25.540,0:00:28.260 means is, you give me an x. 0:00:28.260,0:00:31.660 You can imagine this function[br]is kind of eating up this x. 0:00:31.660,0:00:34.190 You pop an x into[br]this function. 0:00:34.190,0:00:36.480 This function is just[br]a set of rules. 0:00:36.480,0:00:39.150 It's going to say, oh,[br]with that x, I 0:00:39.150,0:00:41.230 associate some value y. 0:00:41.230,0:00:42.945 You can imagine it is[br]some type of a box. 0:00:45.900,0:00:47.990 That is a function. 0:00:47.990,0:00:53.830 When I give it some number[br]x, it'll give me some 0:00:53.830,0:00:56.990 other number y. 0:00:56.990,0:00:58.160 This might seem a[br]little abstract. 0:00:58.160,0:00:59.360 What are these x's and y's? 0:00:59.360,0:01:02.830 Maybe I have a function-- let[br]me make it like this. 0:01:02.830,0:01:04.190 Let's say I have a function[br]definition 0:01:04.190,0:01:05.720 that looks like this. 0:01:05.720,0:01:11.770 For any x you give me, I'm going[br]to produce 1 if x is 0:01:11.770,0:01:14.440 equal to-- I don't know-- 0. 0:01:14.440,0:01:18.730 I'm going to produce 2[br]if x is equal to 1. 0:01:18.730,0:01:21.320 And I'm going to produce[br]3 otherwise. 0:01:24.790,0:01:28.720 So now we've defined what's[br]going on inside of the box. 0:01:28.720,0:01:31.630 So let's draw the[br]box around it. 0:01:31.630,0:01:33.650 This is our box. 0:01:33.650,0:01:35.940 This is just an arbitrary[br]function definition, but 0:01:35.940,0:01:37.760 hopefully it'll help you[br]understand what's actually 0:01:37.760,0:01:40.070 going on with a function. 0:01:40.070,0:01:47.500 So now if I make x is equal to--[br]if I pick x is equal to 0:01:47.500,0:01:52.480 7, now what is f of x going[br]to be equal to? 0:01:52.480,0:01:56.400 What is f of 7 going[br]to be equal to? 0:01:56.400,0:01:58.020 So I take 7 into the box. 0:01:58.020,0:01:59.700 You could view it as some[br]type of a computer. 0:01:59.700,0:02:02.770 The computer looks at that x and[br]then looks at its rules. 0:02:02.770,0:02:04.060 It says, OK. x is 7. 0:02:04.060,0:02:06.270 Well x isn't 0. x isn't 1. 0:02:06.270,0:02:08.229 I go to the otherwise[br]situation. 0:02:08.229,0:02:10.100 So I'm going to pop out a 3. 0:02:10.100,0:02:12.040 So f of 7 is equal to 3. 0:02:12.040,0:02:15.320 So we write f of 7[br]is equal to 3. 0:02:15.320,0:02:18.760 Where f is the name of this[br]function, this rule system, or 0:02:18.760,0:02:21.310 this association, this[br]mapping, whatever you 0:02:21.310,0:02:22.190 want to call it. 0:02:22.190,0:02:24.350 When you give it a 7,[br]it'll produce a 3. 0:02:24.350,0:02:27.460 When you give f a 7,[br]it'll produce a 3. 0:02:27.460,0:02:31.240 What is f of 2? 0:02:31.240,0:02:34.690 Well, that means instead of x[br]is equal to 7, I'm going to 0:02:34.690,0:02:36.420 give it an x equal 2. 0:02:36.420,0:02:38.550 Then the little computer inside[br]the function is going 0:02:38.550,0:02:42.550 to say, OK, let's see,[br]when x is equal to 2. 0:02:42.550,0:02:44.410 No, I'm still in the otherwise[br]situation. 0:02:44.410,0:02:45.910 x isn't 0 or 1. 0:02:45.910,0:02:50.800 So once again f of[br]x is equal to 3. 0:02:53.470,0:02:56.970 So, this is f of 2 is[br]also equal to 3. 0:02:56.970,0:03:03.200 Now what happens if x[br]is now equal to 1? 0:03:03.200,0:03:05.100 Well then it's just going[br]to turn over this. 0:03:05.100,0:03:07.990 So f of 1. 0:03:07.990,0:03:10.080 It's going to look at its[br]rules right here. 0:03:10.080,0:03:11.620 Oh look, x is equal to 1. 0:03:11.620,0:03:13.350 I can use my rule right here. 0:03:13.350,0:03:15.520 So when x is equal to[br]1, I spit out a 2. 0:03:15.520,0:03:18.750 So f of 1 is going[br]to be equal to 2. 0:03:18.750,0:03:22.290 I spit out f of 1, which is[br]equal to 2 in that situation. 0:03:22.290,0:03:24.420 That's all a function is. 0:03:24.420,0:03:29.120 Now, with that in mind, let's[br]do some of these example 0:03:29.120,0:03:31.620 problems. They tell us for[br]each of the following 0:03:31.620,0:03:35.010 functions, evaluate these[br]different functions-- these 0:03:35.010,0:03:37.570 are the different boxes they've[br]created-- at these 0:03:37.570,0:03:39.070 different points. 0:03:39.070,0:03:42.800 Let's do part a first. They're[br]defining the box. 0:03:42.800,0:03:47.880 f of x is equal to negative[br]2x plus 3. 0:03:47.880,0:03:51.790 They want to know what happens[br]when f is equal to negative 3. 0:03:51.790,0:03:54.300 Well f is equal to negative 3,[br]this is telling me what do I 0:03:54.300,0:03:55.430 do with the x? 0:03:55.430,0:03:57.110 What do I produce? 0:03:57.110,0:04:00.060 Wherever I see an x, I replace[br]it with the negative 3. 0:04:00.060,0:04:02.060 So it's going to be equal[br]to negative 2. 0:04:02.060,0:04:04.780 Let me do it this way, so you[br]see exactly what I'm doing. 0:04:04.780,0:04:06.520 That negative 3, I'll do[br]it in that bold color. 0:04:06.520,0:04:13.130 It's negative 2 times[br]negative 3 plus 3. 0:04:13.130,0:04:16.149 Notice wherever there was an[br]x, I put the negative 3. 0:04:16.149,0:04:19.250 So I know what the black box[br]is going to produce. 0:04:19.250,0:04:21.600 This is going to be equal to[br]negative 2 times negative 3 is 0:04:21.600,0:04:25.640 6 plus 3, which is equal to 9. 0:04:25.640,0:04:29.470 So f of negative 3[br]is equal to 9. 0:04:29.470,0:04:32.130 What about f of 7? 0:04:32.130,0:04:36.340 I'll do the same thing one more[br]time. f of-- I'll do 7 in 0:04:36.340,0:04:43.120 yellow-- f of 7 is going to[br]be equal to negative 2 0:04:43.120,0:04:47.650 times 7 plus 3. 0:04:50.480,0:04:55.140 So this is equal to negative 14[br]plus 3, which is equal to 0:04:55.140,0:04:57.260 negative 11. 0:04:57.260,0:05:03.940 You put in-- let me make it very[br]clear-- you put in a 7 0:05:03.940,0:05:11.060 into our function f here and it[br]will pop out a negative 11. 0:05:11.060,0:05:13.310 That's what this just[br]told us right there. 0:05:13.310,0:05:14.760 This is the rule. 0:05:14.760,0:05:18.470 This is completely analogous[br]to what I did up here. 0:05:18.470,0:05:20.980 This is the rule of[br]our function. 0:05:20.980,0:05:24.430 Let's do the next two. 0:05:24.430,0:05:25.200 I won't do part b. 0:05:25.200,0:05:26.330 You can do part b for fun. 0:05:26.330,0:05:29.650 I'll do part c after that, just[br]for the sake of time. 0:05:29.650,0:05:32.540 Now we are at f of 0. 0:05:32.540,0:05:33.810 Here I'll just do[br]it in one color. 0:05:33.810,0:05:35.300 I think you're getting[br]the idea. f of 0. 0:05:35.300,0:05:37.500 Wherever we see an[br]x, we put a 0. 0:05:37.500,0:05:40.005 So negative 2 times 0 plus 3. 0:05:43.100,0:05:44.345 Well, that's just[br]going to be a 0. 0:05:44.345,0:05:47.300 So f of 0 is 3. 0:05:47.300,0:05:49.000 Then one last one. f of z. 0:05:49.000,0:05:51.720 They want to keep it[br]abstract for us. 0:05:51.720,0:05:52.780 Here I'll color code it. 0:05:52.780,0:05:55.800 So f of z. 0:05:55.800,0:05:59.150 Let me make the z in[br]a different color. 0:05:59.150,0:06:00.900 f of z. 0:06:00.900,0:06:06.210 Everywhere that we saw[br]an x, we will now 0:06:06.210,0:06:07.750 replace it with a z. 0:06:07.750,0:06:09.240 Negative 2. 0:06:09.240,0:06:12.040 Instead of an x, we're going[br]to put a z there. 0:06:12.040,0:06:13.860 We're going to put an[br]orange z there. 0:06:13.860,0:06:19.760 Negative 2 times z plus 3. 0:06:19.760,0:06:24.330 And that's our answer. f of[br]z is negative 2z plus 3. 0:06:24.330,0:06:28.110 If you imagine our box,[br]the function f. 0:06:28.110,0:06:38.130 You put in a z, you are going to[br]get out a negative 2 times 0:06:38.130,0:06:43.480 whatever that z is plus 3. 0:06:43.480,0:06:44.520 That's all this is saying. 0:06:44.520,0:06:47.830 It's a little bit more abstract,[br]but same exact idea. 0:06:47.830,0:06:52.030 Now let's just do part c here. 0:06:52.030,0:06:53.330 Let me clear this actually. 0:06:53.330,0:06:55.820 I'm running out of[br]real estate. 0:06:55.820,0:06:59.102 Let me clear all of[br]this business. 0:06:59.102,0:07:02.910 Let me clear all of[br]this business. 0:07:02.910,0:07:03.810 We can do part c. 0:07:03.810,0:07:05.370 I'm skipping part b. 0:07:05.370,0:07:07.710 You can work on that part. 0:07:07.710,0:07:10.830 Part b. 0:07:10.830,0:07:13.430 They tell us-- this is our[br]function definition. 0:07:13.430,0:07:16.680 Sorry, I said I was[br]doing part c. 0:07:16.680,0:07:18.610 This is our function[br]definition. 0:07:18.610,0:07:26.300 f of x is equal to 5 times[br]2 minus x over 11. 0:07:26.300,0:07:29.440 So let's apply these different[br]values of x, these different 0:07:29.440,0:07:32.620 inputs into our function. 0:07:32.620,0:07:39.900 So f of negative 3 is equal to[br]5 times 2 minus-- wherever we 0:07:39.900,0:07:42.250 see an x, we put a negative 3. 0:07:42.250,0:07:45.620 2 minus negative 3 over 11. 0:07:45.620,0:07:48.700 This is equal to 2 plus 3. 0:07:48.700,0:07:50.870 This is equal to 5. 0:07:50.870,0:07:53.260 So you get 5 times 5 over 11. 0:07:53.260,0:07:57.120 That's equal to 25/11. 0:07:57.120,0:07:57.850 Let's do this one. 0:07:57.850,0:07:59.990 F of 7. 0:07:59.990,0:08:06.680 For this second function right[br]here, f of 7 is equal to 5 0:08:06.680,0:08:11.160 times 2 minus-- now[br]for x we have a 7. 0:08:11.160,0:08:14.360 2 minus 7 over 11. 0:08:14.360,0:08:15.540 So what is this going[br]to be equal to? 0:08:15.540,0:08:18.250 2 minus 7 is negative 5. 0:08:18.250,0:08:23.780 5 times negative 5 is[br]negative 25/11. 0:08:23.780,0:08:27.410 Then finally, well we have[br]two more. f of 0. 0:08:27.410,0:08:35.000 That's equal to 5 times 2 minus[br]0 So this is just 2. 0:08:35.000,0:08:36.130 5 times 2 is 10. 0:08:36.130,0:08:38.850 So this is equal to 10/11. 0:08:38.850,0:08:39.840 One more. 0:08:39.840,0:08:42.059 f of z. 0:08:42.059,0:08:43.299 Well every time we saw[br]an x, we're going to 0:08:43.299,0:08:44.490 replace it with a z. 0:08:44.490,0:08:49.960 It's equal to 5 times[br]2 minus z over 11. 0:08:49.960,0:08:50.630 And that's our answer. 0:08:50.630,0:08:51.910 We could distribute the 5. 0:08:51.910,0:08:57.210 You could say this is the same[br]thing as 10 minus 5z over 11. 0:08:57.210,0:09:00.260 We could even write it in[br]slope-intercept form. 0:09:00.260,0:09:06.000 This is the same thing as[br]minus 5/11 z plus 10/11. 0:09:06.000,0:09:06.990 These are all equivalent. 0:09:06.990,0:09:10.430 But that is what f[br]of z is equal to. 0:09:10.430,0:09:11.590 Now. 0:09:11.590,0:09:15.510 A function, we said, if you give[br]me any x value, I will 0:09:15.510,0:09:16.470 give you an output. 0:09:16.470,0:09:19.120 I will give you an f of x. 0:09:19.120,0:09:23.040 So if this is our function,[br]you give me an x, it will 0:09:23.040,0:09:26.550 produce an f of x. 0:09:26.550,0:09:29.680 It can only produce 1[br]f of x for any x. 0:09:29.680,0:09:32.840 You can't have a function that[br]could produce two possible 0:09:32.840,0:09:34.700 values for an x. 0:09:34.700,0:09:37.540 So you can't have a function--[br]this would be an invalid 0:09:37.540,0:09:42.790 function definition-- f[br]of x is equal to 3 if 0:09:42.790,0:09:45.230 x is equal to 0. 0:09:45.230,0:09:49.240 Or it could be equal to[br]4 if x is equal to 0. 0:09:49.240,0:09:53.170 Because in this situation, we[br]don't know what f of 0 is. 0:09:53.170,0:09:54.090 What it's going to be equal? 0:09:54.090,0:09:56.330 It says if x is equal to 0, it[br]should be 3 or it could be-- 0:09:56.330,0:09:57.310 we don't know. 0:09:57.310,0:09:57.830 We don't know. 0:09:57.830,0:09:58.190 We don't know. 0:09:58.190,0:10:01.550 This is not a function[br]even though it might 0:10:01.550,0:10:02.800 have looked like one. 0:10:07.700,0:10:12.250 So you can't have two f of[br]x values for one x value. 0:10:12.250,0:10:16.020 So let's see which of these[br]graphs are functions. 0:10:16.020,0:10:18.390 To figure that out, you could[br]say, look at any x value 0:10:18.390,0:10:21.850 here-- pick any x value-- I have[br]exactly one f of x value. 0:10:21.850,0:10:25.090 This is y is equal to[br]f of x right here. 0:10:25.090,0:10:28.950 I have exactly only one--[br]at that x, that 0:10:28.950,0:10:30.550 is my y value here. 0:10:30.550,0:10:32.970 So you could have a vertical[br]line test, which says at any 0:10:32.970,0:10:35.720 point if you draw a vertical[br]line-- notice a vertical line 0:10:35.720,0:10:37.570 is for a certain x value. 0:10:37.570,0:10:41.920 That shows that I only have[br]one y value at that point. 0:10:41.920,0:10:43.630 So this is a valid function. 0:10:43.630,0:10:46.240 Any time you draw a vertical[br]line, it will only intersect 0:10:46.240,0:10:47.610 the graph once. 0:10:47.610,0:10:50.410 So this is a valid function. 0:10:50.410,0:10:52.220 Now what about this[br]one right here? 0:10:52.220,0:10:53.960 I could draw a vertical[br]line, let's say, at 0:10:53.960,0:10:55.230 that point right there. 0:10:55.230,0:10:58.650 For that x, this relation[br]seems to have two 0:10:58.650,0:11:00.860 possible f of x's. 0:11:00.860,0:11:04.550 f of x could be that value or[br]f of x could be that value. 0:11:04.550,0:11:05.270 Right? 0:11:05.270,0:11:07.520 We're intersecting[br]the graph twice. 0:11:07.520,0:11:08.840 So this is not a function. 0:11:08.840,0:11:11.150 We're doing exactly what[br]I described here. 0:11:11.150,0:11:15.090 For a certain x, we're[br]describing two possible y's 0:11:15.090,0:11:16.800 that could be equal to f of x. 0:11:16.800,0:11:19.220 So this is not a function. 0:11:19.220,0:11:20.830 Over here, same thing. 0:11:20.830,0:11:22.310 You draw a vertical[br]line right there. 0:11:22.310,0:11:24.540 You're intersecting[br]the graph twice. 0:11:24.540,0:11:26.000 This is not a function. 0:11:26.000,0:11:30.590 You're defining two possible[br]y values for 1 x value. 0:11:30.590,0:11:31.490 Let's go to this function. 0:11:31.490,0:11:33.160 It's kind of a weird[br]looking function. 0:11:33.160,0:11:34.750 Looks like a reversed[br]check mark. 0:11:34.750,0:11:37.020 But any time you draw a vertical[br]line, you're only 0:11:37.020,0:11:38.720 intersecting it once. 0:11:38.720,0:11:40.420 So this is a valid function. 0:11:40.420,0:11:43.470 For every x, you only have[br]one y associated. 0:11:43.470,0:11:46.450 Or only one f of x associated[br]with it. 0:11:46.450,0:11:48.960 Anyway, hopefully you[br]found that useful.