1 00:00:04,830 --> 00:00:07,140 Robert McKeown: Hello and welcome to ALEKS 2 00:00:07,140 --> 00:00:10,230 walkthrough video number five. My name is Robert 3 00:00:10,230 --> 00:00:11,910 McKeown and today we're going to be talking 4 00:00:11,910 --> 00:00:16,830 about functions and graphs, students. In my 5 00:00:16,830 --> 00:00:19,530 experience at York University, and I'm not been 6 00:00:19,590 --> 00:00:22,620 here very long, students find this topic 7 00:00:22,620 --> 00:00:26,070 extremely challenging. I don't think it's 8 00:00:26,070 --> 00:00:28,620 particularly hard. But what you have to do is 9 00:00:28,620 --> 00:00:31,590 you have to memorize a number of definitions, 10 00:00:31,800 --> 00:00:34,530 and then apply those definitions to different 11 00:00:34,530 --> 00:00:37,710 situations. And that can be a little bit 12 00:00:37,710 --> 00:00:39,210 difficult because you actually have to learn the 13 00:00:39,210 --> 00:00:41,340 definitions, you actually have to do the work 14 00:00:41,430 --> 00:00:44,190 doesn't matter how smart you are smart is not 15 00:00:44,190 --> 00:00:46,710 going to help you if you don't have the 16 00:00:46,710 --> 00:00:50,070 knowledge necessary to answer the question. And 17 00:00:50,070 --> 00:00:54,030 so this is actually the longest series of videos 18 00:00:54,060 --> 00:00:57,120 of all a topics because there's just so much 19 00:00:58,470 --> 00:01:00,960 that I want to explain to you. And then you just 20 00:01:00,960 --> 00:01:04,290 need to know. So you can think of maybe the 21 00:01:04,290 --> 00:01:08,520 first four topics as being rather introductory, 22 00:01:08,610 --> 00:01:10,590 didn't require a lot of background information 23 00:01:10,590 --> 00:01:13,650 or background knowledge, background knowledge, 24 00:01:13,800 --> 00:01:16,500 now we're building off the knowledge you learned 25 00:01:16,500 --> 00:01:20,340 in topics 123, and four, and we're going forward 26 00:01:20,340 --> 00:01:25,110 into more advanced mathematics and into economic 27 00:01:25,110 --> 00:01:30,510 applications. So again, before we get started, 28 00:01:30,540 --> 00:01:35,130 make sure you have a pencil, a pen, paper, or if 29 00:01:35,130 --> 00:01:38,220 you're really fancy, get yourself a tablet with 30 00:01:38,220 --> 00:01:40,770 a writing pen. That's a really great tool to 31 00:01:40,770 --> 00:01:43,350 have. I was so impressed with the my students 32 00:01:43,350 --> 00:01:48,000 who had one that I went out and bought one. And 33 00:01:48,000 --> 00:01:49,710 I use it in class. If you've ever had a lecture 34 00:01:49,710 --> 00:01:52,830 with me, you'll see it for sure. So without any 35 00:01:52,830 --> 00:01:58,170 further ado, let's get down to the problems. To 36 00:01:58,170 --> 00:02:01,290 answer the topic five questions correctly, you 37 00:02:01,290 --> 00:02:03,870 have to understand a few definitions. And our 38 00:02:03,870 --> 00:02:08,190 first definition is what is a function. So you 39 00:02:08,190 --> 00:02:11,940 can read a formal definition on the slides. One 40 00:02:11,940 --> 00:02:13,800 thing I'd like you to think about is I'd like 41 00:02:13,800 --> 00:02:18,870 you to think about a streetlight. So functions 42 00:02:19,200 --> 00:02:23,040 don't have to be mathematical in nature, we 43 00:02:23,040 --> 00:02:26,850 could have an X variable, which is the 44 00:02:26,850 --> 00:02:32,850 streetlight can be a red light, can be red. Or 45 00:02:32,850 --> 00:02:33,900 it can be a green light. 46 00:02:39,090 --> 00:02:44,730 And there's a function that transforms that 47 00:02:44,790 --> 00:02:51,090 observing that light and to an outcome, which is 48 00:02:51,150 --> 00:02:55,890 stop your car forever the red light, and if it's 49 00:02:55,890 --> 00:03:02,250 a green light, it's go. And so we've got two 50 00:03:02,250 --> 00:03:06,600 ordered pairs here, we've got an ordered pair 51 00:03:06,600 --> 00:03:14,280 that's red light, and associated with stop and 52 00:03:14,280 --> 00:03:22,260 another ordered pair, that's a green light. And 53 00:03:22,860 --> 00:03:28,230 it leads to go. So that's an example of a 54 00:03:28,230 --> 00:03:32,400 function in everyday life. We're gonna focus on 55 00:03:32,400 --> 00:03:36,030 numerical functions here. But in economics, 56 00:03:36,030 --> 00:03:38,880 sometimes we don't want to use numbers, we want 57 00:03:38,880 --> 00:03:44,940 to use concepts like this, and we do so here is 58 00:03:44,940 --> 00:03:48,180 another definition, maybe a simpler definition 59 00:03:48,180 --> 00:03:53,400 of what a function is. So we've got a function 60 00:03:53,400 --> 00:04:01,350 something like this. And we might have y is 61 00:04:01,350 --> 00:04:09,570 equal to some function of x. So let's consider a 62 00:04:09,570 --> 00:04:20,850 function x squared. So if x is equal to zero, y 63 00:04:20,880 --> 00:04:26,370 is going to also be equal to zero. And if x is 64 00:04:26,370 --> 00:04:33,870 equal to one, y will be equal to one. If x is 65 00:04:33,870 --> 00:04:38,490 equal to negative one, y is also equal to one. 66 00:04:39,390 --> 00:04:43,110 And so I'll tell you that x squared is a 67 00:04:43,110 --> 00:05:07,020 function for every x there is one Y value. Now I 68 00:05:07,020 --> 00:05:09,330 haven't proved it. But if you want, you can go 69 00:05:09,330 --> 00:05:11,280 home, you can put in any number you want into 70 00:05:11,280 --> 00:05:15,510 that. And you're only going to get one y value, 71 00:05:15,630 --> 00:05:17,730 whatever number you put into that function, 72 00:05:18,330 --> 00:05:20,790 you're going to get only one y value. In fact, 73 00:05:20,790 --> 00:05:23,520 that's what makes it a function. Let me show you 74 00:05:23,520 --> 00:05:26,040 an example of something that is not a function. 75 00:05:26,580 --> 00:05:38,310 So this is a function. What if I have? I can't 76 00:05:38,310 --> 00:05:41,250 use that terminology, what if I have y squared 77 00:05:42,030 --> 00:05:50,700 plus x squared is equal to 25? Well, I've got y 78 00:05:50,700 --> 00:05:54,210 is equal to a function of x. So why don't I 79 00:05:54,210 --> 00:05:58,350 rearrange this thing over here and isolate y on 80 00:05:58,350 --> 00:06:03,600 one side, so that it looks like the function I 81 00:06:03,600 --> 00:06:05,640 showed you on the left hand side of the screen. 82 00:06:06,060 --> 00:06:10,110 So I can take, I know that y squared is going to 83 00:06:10,110 --> 00:06:18,480 be equal to 25 minus x squared, I can write y, 84 00:06:19,650 --> 00:06:21,600 I'll take the square root of both sides of the 85 00:06:21,600 --> 00:06:33,780 expression. So I've got 25 minus x squared. But 86 00:06:33,990 --> 00:06:39,090 notice that this is going to be plus or minus, 87 00:06:39,630 --> 00:06:41,760 whenever I take the square root of something 88 00:06:41,760 --> 00:06:45,270 that's been squared, it could be positive or 89 00:06:45,270 --> 00:06:52,710 negative. And so with this example, for each x 90 00:06:53,670 --> 00:07:07,080 value, there are two possible I'm not going to 91 00:07:07,080 --> 00:07:08,790 say each, let's just say for some, 92 00:07:15,330 --> 00:07:23,760 there are two possible y values. So it's not a 93 00:07:23,760 --> 00:07:26,100 function, the language we like to use in 94 00:07:26,100 --> 00:07:30,120 mathematics and economics is that the x values 95 00:07:30,300 --> 00:07:33,270 are in something called the domain. And the y 96 00:07:33,270 --> 00:07:36,720 values are in something called the range. So if 97 00:07:36,720 --> 00:07:41,250 I've got a set of axes here, and maybe I've got 98 00:07:41,250 --> 00:07:47,190 one, or 01, and two, here, this is the domain. 99 00:07:49,500 --> 00:07:51,270 And then there's some function 100 00:07:56,850 --> 00:07:58,350 that takes us to, 101 00:08:08,190 --> 00:08:11,280 I'm gonna just change this one to a negative 102 00:08:11,280 --> 00:08:16,590 one, like our previous example. And negative one 103 00:08:16,590 --> 00:08:23,070 also maps to Y equals one. So the function is 104 00:08:23,070 --> 00:08:27,150 mapping from the domain into the range. And if 105 00:08:27,150 --> 00:08:29,730 you want to read more about it, I found a neat 106 00:08:30,570 --> 00:08:34,260 website down here that you can read more about 107 00:08:34,860 --> 00:08:37,980 domain and range and what a function is what a 108 00:08:37,980 --> 00:08:43,500 function is not. There is a special kind of 109 00:08:43,500 --> 00:08:46,320 function, and it's called a one to one function. 110 00:08:47,190 --> 00:08:51,210 Now, when we had this previous example, we had a 111 00:08:51,210 --> 00:08:54,930 function that was equal to x squared. But you 112 00:08:54,930 --> 00:09:00,180 can see here that, and I'll get rid of that 113 00:09:00,180 --> 00:09:03,450 because we don't care about it. But you can see 114 00:09:03,450 --> 00:09:08,040 that there's more than one x value factor or two 115 00:09:08,340 --> 00:09:19,650 that lead to one y value. So this is not a one 116 00:09:20,790 --> 00:09:23,730 to one function. 117 00:09:30,330 --> 00:09:31,380 Now let me see. 118 00:09:39,330 --> 00:09:45,990 I could write a one to one function, 119 00:09:53,910 --> 00:10:01,860 which is just f of x is equal to x And notice 120 00:10:01,860 --> 00:10:13,830 that each number in the domain leads to a unique 121 00:10:17,190 --> 00:10:19,320 number in the range. 122 00:10:26,100 --> 00:10:39,510 So f of x equal to x is a one to one function. 123 00:10:44,070 --> 00:10:47,970 So there's, this is a function over here, but 124 00:10:47,970 --> 00:10:50,610 it's not a one to one function. 125 00:10:57,059 --> 00:10:59,339 It's a function but not a one to one function 126 00:10:59,369 --> 00:11:05,639 over here. This is a function and it is a one to 127 00:11:05,639 --> 00:11:06,509 one function. 128 00:11:12,899 --> 00:11:15,389 One to One functions are important because every 129 00:11:15,389 --> 00:11:19,319 one to one function has an inverse function has 130 00:11:19,319 --> 00:11:21,869 an inverse function. So let's consider another 131 00:11:21,869 --> 00:11:27,479 one to one function, I'll have 01 and two in my 132 00:11:27,479 --> 00:11:31,529 domain. And this time, I'll just say let the 133 00:11:31,529 --> 00:11:38,039 function be equal to two x. And there's my 134 00:11:38,039 --> 00:11:44,759 domain. And I'm going to have Y values here, if 135 00:11:44,759 --> 00:11:50,339 x is zero, y is zero. If x is one, y is two, and 136 00:11:50,339 --> 00:11:51,359 if x is two, 137 00:11:57,120 --> 00:12:02,100 if x is two, y is four. Now let's consider the 138 00:12:02,100 --> 00:12:06,840 inverse. So if we have the inverse of this 139 00:12:06,840 --> 00:12:11,310 function, this functions unique inverse, then 140 00:12:17,610 --> 00:12:22,320 it's going to reverse, essentially just going to 141 00:12:22,320 --> 00:12:23,460 reverse the mapping. 142 00:12:33,419 --> 00:12:38,969 And the inverse of this function is equal to one 143 00:12:38,969 --> 00:12:40,919 half times x. 144 00:12:46,860 --> 00:12:49,200 So inverse functions are not very, very 145 00:12:49,200 --> 00:12:53,100 complicated, in their fundamental or simplest 146 00:12:53,100 --> 00:12:56,700 form, they're just reversing. Where we're moving 147 00:12:56,700 --> 00:12:59,250 from, are we moving from X to get y or are we 148 00:12:59,250 --> 00:13:04,140 moving? Are we taking y to get X inverse 149 00:13:04,140 --> 00:13:08,640 functions have a very interesting property. So 150 00:13:08,700 --> 00:13:16,740 this little operator here creates a composite. 151 00:13:17,400 --> 00:13:21,450 So this is a to operator to create a composite 152 00:13:22,200 --> 00:13:27,480 function. And more commonly, it's written like 153 00:13:27,480 --> 00:13:32,010 this, which is basically to say that we have the 154 00:13:32,040 --> 00:13:40,050 inverse function and we plug it into the 155 00:13:40,050 --> 00:13:45,990 original function. We're going to end up with 156 00:13:46,920 --> 00:13:51,060 just x whatever that x value is. So let me give 157 00:13:51,060 --> 00:13:54,480 you an example from the previous one, we had f 158 00:13:54,480 --> 00:14:00,270 of x is equal to two the inverse of that 159 00:14:00,270 --> 00:14:04,020 function or excuse me, that was two x, the 160 00:14:04,050 --> 00:14:14,040 inverse was equal to one half x. and say I 161 00:14:14,040 --> 00:14:16,350 wanted to evaluate 162 00:14:49,380 --> 00:14:51,690 we always start with the inner right whatever is 163 00:14:51,690 --> 00:14:54,060 in the brackets is the first order of 164 00:14:54,060 --> 00:14:59,370 operations. So F inverse three 165 00:15:08,250 --> 00:15:09,570 We have one half 166 00:15:16,260 --> 00:15:20,100 times three, which gives us 167 00:15:27,000 --> 00:15:30,540 three over two. Now let's take that three over 168 00:15:30,540 --> 00:15:42,270 two and plug it into the original function. And 169 00:15:42,270 --> 00:15:52,680 we get two times three over two, which is equal 170 00:15:52,680 --> 00:15:57,570 to three. And that is exactly what this special 171 00:15:57,570 --> 00:16:01,860 property is all about. And it says that if we 172 00:16:01,860 --> 00:16:06,900 take the inverse of x, and then we take that 173 00:16:06,900 --> 00:16:08,940 inverse of x and take the function of the whole 174 00:16:08,940 --> 00:16:13,290 thing, we get x. We've gone through an 175 00:16:13,290 --> 00:16:15,870 introduction together, let's apply what we've 176 00:16:15,870 --> 00:16:20,070 learned to answer the next three questions. So 177 00:16:20,070 --> 00:16:23,820 we're given two functions. The first function g 178 00:16:25,380 --> 00:16:28,890 doesn't have any calculations, it's just a set 179 00:16:28,890 --> 00:16:33,450 of ordered pairs. So given us the ordered pairs 180 00:16:33,480 --> 00:16:36,060 directly, we don't have to do any calculation to 181 00:16:36,060 --> 00:16:40,980 find them. The second function age is a function 182 00:16:40,980 --> 00:16:42,480 as you're used to seeing it, where we're going 183 00:16:42,480 --> 00:16:45,810 to have to perform some calculations. Now the 184 00:16:45,810 --> 00:16:50,460 first question is asking us to find the inverse 185 00:16:51,750 --> 00:17:02,430 of G. So remember that if we have x, mapping 186 00:17:02,430 --> 00:17:09,720 into y, according to G, now we've got the 187 00:17:09,720 --> 00:17:16,170 inverse of G where we're mapping from y to x. So 188 00:17:16,170 --> 00:17:21,000 g inverse is going to take the y value here, 189 00:17:22,500 --> 00:17:25,950 move it to the first position where the domain 190 00:17:25,950 --> 00:17:31,350 is. And if y is equal to seven, we're going to 191 00:17:31,350 --> 00:17:35,820 map to x equal to negative five. And we're going 192 00:17:35,820 --> 00:17:38,010 to do that for each of the ordered pairs. 193 00:17:48,930 --> 00:17:51,690 And the squirrely back, it represents a set 194 00:17:55,710 --> 00:17:58,320 beginning and the ending of the set. And it's a 195 00:17:58,320 --> 00:18:03,360 set of ordered pairs. And we're done. So finding 196 00:18:03,360 --> 00:18:07,110 the inverse function, when you're given a set of 197 00:18:07,110 --> 00:18:10,170 ordered pairs, is very straightforward, you just 198 00:18:10,170 --> 00:18:12,750 switch the positions of the domain and the 199 00:18:12,750 --> 00:18:19,260 range. So we're asked to find the inverse of h 200 00:18:19,260 --> 00:18:23,220 of x. So I'm going to start it off by writing y 201 00:18:23,610 --> 00:18:29,610 is equal to three x plus 13. And now I'm going 202 00:18:29,610 --> 00:18:36,180 to solve for x. So I'm going to isolate x on the 203 00:18:36,180 --> 00:18:39,450 left hand side of the equation. So I'm going to 204 00:18:39,450 --> 00:18:45,840 have three x is equal to y minus 13. And then x 205 00:18:45,840 --> 00:18:52,200 is equal to y minus 13 divided by three, then 206 00:18:52,200 --> 00:19:04,740 I'm going to replace x with y, and y with x. So 207 00:19:04,740 --> 00:19:09,600 I'm going to have y is equal to x minus 13 over 208 00:19:09,600 --> 00:19:12,930 three, so the inverse 209 00:19:18,600 --> 00:19:23,760 is equal to x minus 13 over three, so that's the 210 00:19:23,790 --> 00:19:26,940 inverse of the function. So it started switching 211 00:19:26,940 --> 00:19:29,760 the x's and y's around. But that's just because 212 00:19:29,760 --> 00:19:32,490 we want to show the inverse function should be a 213 00:19:32,490 --> 00:19:39,240 function of x. Our next question is asking for 214 00:19:39,240 --> 00:19:42,660 the composite function h composite with its 215 00:19:42,690 --> 00:19:46,920 inverse. And it wants to wants us to evaluate 216 00:19:46,920 --> 00:19:50,160 that composite function when x is equal to four. 217 00:19:50,640 --> 00:19:53,820 So why don't I do this the hard way, and then we 218 00:19:53,820 --> 00:19:56,490 can I can show you and remind you how to do it 219 00:19:56,490 --> 00:20:00,690 the easy way. So another way of writing In this 220 00:20:02,100 --> 00:20:03,840 is to say that we have H 221 00:20:10,710 --> 00:20:13,530 inverse of four, and then that that inverse is 222 00:20:13,530 --> 00:20:17,640 actually within the original function. I've 223 00:20:18,000 --> 00:20:20,250 taken the inverse function we came up with on 224 00:20:20,250 --> 00:20:25,410 the previous slide, and brought it with me. And 225 00:20:25,410 --> 00:20:30,030 so why don't we do that one first. So H, one of 226 00:20:30,030 --> 00:20:34,050 four is going to be equal to four minus 13 227 00:20:35,100 --> 00:20:37,650 divided by three, which is equal to negative 228 00:20:37,650 --> 00:20:43,350 nine over three, which is equal to negative 229 00:20:43,350 --> 00:20:49,590 three. Now I'm going to use, I'm going to plug 230 00:20:49,590 --> 00:20:51,930 that negative three into the original function 231 00:20:55,740 --> 00:21:02,550 through three times negative three, plus 13 is 232 00:21:02,550 --> 00:21:07,350 equal to negative nine plus 13, which is equal 233 00:21:07,350 --> 00:21:11,970 to four. Right? And that's exactly what we 234 00:21:11,970 --> 00:21:19,710 should get because we know that the composite of 235 00:21:19,800 --> 00:21:26,220 the H function and its inverse I'll delete that 236 00:21:26,220 --> 00:21:28,650 little x when it's evaluated at four whether 237 00:21:28,650 --> 00:21:32,280 it's x is going to be equal to four. Right? Just 238 00:21:32,280 --> 00:21:36,990 like if we didn't have a number to evaluated at, 239 00:21:37,920 --> 00:21:39,780 we would get our x value