1 00:00:00,989 --> 00:00:05,652 [MUSIC] So, before we talk about the domain of the square root function, we 2 00:00:05,652 --> 00:00:10,316 just want to remind ourselves what the square root function even is. 3 00:00:10,316 --> 00:00:14,131 So here, I've made a graph of the square root function. 4 00:00:14,131 --> 00:00:19,926 And along the x-axis, I plot the numbers one to sixteen and in the y-axis I've got 5 00:00:19,926 --> 00:00:24,730 the numbers one through four. And then in this green curve here, I've 6 00:00:24,730 --> 00:00:27,840 plotted the the, the square root function. 7 00:00:27,840 --> 00:00:31,316 What is the square root, right? Well, here's an example. 8 00:00:31,316 --> 00:00:36,563 Here, I've got the square root of four. And I'm saying the square root of four is 9 00:00:36,563 --> 00:00:39,448 two. What that means, is if I take the number 10 00:00:39,448 --> 00:00:44,402 two and I square it, I get back four. I don't know, if I move over to, say, the 11 00:00:44,402 --> 00:00:49,255 square root of nine, I get three. And that's because three squared is nine, 12 00:00:49,255 --> 00:00:52,379 alright, or if I move over a little bit further, 13 00:00:52,379 --> 00:00:57,032 the square root of sixteen is four. And that's because four squared is 14 00:00:57,032 --> 00:01:00,156 sixteen. I know there's some crazier values, too. 15 00:01:00,156 --> 00:01:03,414 If I move over here to the square root of two, 16 00:01:03,414 --> 00:01:08,532 well, the square root of two is this sort of crazy number 1.414213 blah, blah, 17 00:01:08,532 --> 00:01:11,457 blah. And maybe it's a little bit surprising, 18 00:01:11,457 --> 00:01:15,800 that if I take that number and square it, I get back two. 19 00:01:15,800 --> 00:01:20,167 So, what's going on here, alright? The square root function takes a number 20 00:01:20,167 --> 00:01:24,958 and spits out a new number, that new number when you multiply it by itself and 21 00:01:24,958 --> 00:01:27,870 you square it, you get back your original number. 22 00:01:27,870 --> 00:01:32,456 Now, here's the question, what sorts of numbers can I take the square root of? 23 00:01:32,456 --> 00:01:36,862 That's asking the question, what's the domain of the square root function? 24 00:01:36,862 --> 00:01:41,811 Now, that we've seen the graph, let's try to write down in words a definition of 25 00:01:41,811 --> 00:01:45,790 the square root function. So, in light of what we just seen, you 26 00:01:45,790 --> 00:01:51,265 might think that the definition of the function f(x) equals square root of x is 27 00:01:51,265 --> 00:01:55,472 a number which squares to x. There's a problem with this though. 28 00:01:55,472 --> 00:01:59,211 Take a look at say, f(9). What would f(9) be? 29 00:01:59,211 --> 00:02:04,619 Well, if you're thinking the square root of x is a number which squares to x, then 30 00:02:04,619 --> 00:02:08,625 you might think that f(9) would be -3, alright? 31 00:02:08,625 --> 00:02:16,597 Because -3^2 is 9. But then, you might also think that half 32 00:02:16,597 --> 00:02:23,820 of nine should be three, right? Because 3^2 is also 9. 33 00:02:23,820 --> 00:02:27,933 This is bad, alright? A function is supposed to be unambiguous. 34 00:02:27,933 --> 00:02:31,460 It's supposed to have one output for each input. 35 00:02:31,460 --> 00:02:36,479 If you take this as the definition of the square root function, just any number 36 00:02:36,479 --> 00:02:40,482 which squares to x, you've introduced some ambiguity, alright? 37 00:02:40,482 --> 00:02:45,120 What's the square root of nine? Is it -3 or is it +3? 38 00:02:45,120 --> 00:02:49,441 Both of those numbers square to nine. So, this is, this is bad, alright? 39 00:02:49,441 --> 00:02:54,651 The solution is to change the definition. Instead of having the the square root 40 00:02:54,651 --> 00:02:59,290 function be just a number which squares to x, you're going to take it to be the 41 00:02:59,290 --> 00:03:06,672 nonnegative number which squares to x. This is better, alright? 42 00:03:06,672 --> 00:03:13,222 In our example here, if I only am allowed to choose the nonnegative number, which 43 00:03:13,222 --> 00:03:21,147 squares to x, then f(9) equals -3, well, -3 is not nonnegative, -3 is negative. 44 00:03:21,147 --> 00:03:25,029 So, that means that this isn't the case, right? 45 00:03:25,029 --> 00:03:32,226 All I'm left with is f(9) = 3, right? Three is the nonnegative number which 46 00:03:32,226 --> 00:03:35,112 squares to nine. Alright. 47 00:03:35,112 --> 00:03:39,949 So, this will be our definition for for, for the square root function. 48 00:03:39,949 --> 00:03:44,860 The square root of x is the nonnegative number which squares to x. 49 00:03:44,860 --> 00:03:49,410 There's one particular place where this plays out and it's extraordinarily 50 00:03:49,410 --> 00:03:52,020 important. So, let's take a look at that now. 51 00:03:52,020 --> 00:03:56,261 We've got our definition. The squared of x is the nonnegative 52 00:03:56,261 --> 00:04:01,372 number which squares to x. Now, there's one popular misconception 53 00:04:01,372 --> 00:04:06,910 that comes up because of this definition. So, in light of the definition of the 54 00:04:06,910 --> 00:04:09,999 square root, right, the square root of a number being the 55 00:04:09,999 --> 00:04:14,025 nonnegative number which squares the number to the radical, you might be 56 00:04:14,025 --> 00:04:17,390 tricked into thinking that the square root of x squared is x. 57 00:04:17,390 --> 00:04:23,734 That's not true and let's see why. Let's do a specific example where say, x 58 00:04:23,734 --> 00:04:28,240 is -4. So, if I replace the x's here by -4, the 59 00:04:28,240 --> 00:04:32,225 left hand side is the square root of -4 squared, 60 00:04:32,225 --> 00:04:34,770 right? Square root of x squared but with x 61 00:04:34,770 --> 00:04:38,890 replaced with -4. Now, - 4 * -4 is 16. 62 00:04:38,890 --> 00:04:43,980 This is the square root of 16 and the square root of sixteen, the definition of 63 00:04:43,980 --> 00:04:48,100 the square root is the nonnegative number which squares to 16. 64 00:04:48,100 --> 00:04:52,220 There's two numbers that square to 16, +4 and -4. 65 00:04:52,220 --> 00:04:57,250 But the square root is by convention, the nonnegative one, so this is equal to 4. 66 00:04:57,250 --> 00:04:59,410 Duh, look at what happened. 67 00:04:59,410 --> 00:05:05,893 -4, square root of -4^2 + 4, that's the x over here. 68 00:05:05,893 --> 00:05:11,129 This is not true, right. You should not be tricked into thinking 69 00:05:11,129 --> 00:05:15,451 that that's the case. Instead, something else is true, 70 00:05:15,451 --> 00:05:17,695 right? What is true is this. 71 00:05:17,695 --> 00:05:22,350 The square root of x squared is the absolute value of x. 72 00:05:22,350 --> 00:05:26,697 And that works in this specific case, right? 73 00:05:26,697 --> 00:05:33,781 When x is -4, the square root of -4 squared, the square root of 16 is 4. 74 00:05:33,781 --> 00:05:40,924 And 4 really is the absolute value of -4. Alright. So, this is a mistake that comes 75 00:05:40,924 --> 00:05:44,143 up quite a bit. People are often tricked into thinking 76 00:05:44,143 --> 00:05:46,765 that the square root of x squared is just x, 77 00:05:46,765 --> 00:05:50,998 alright? They're just trying to cancel the square roots in the squaring. 78 00:05:50,998 --> 00:05:54,574 That's not possible. Instead, what is true is the square root 79 00:05:54,574 --> 00:05:59,343 of x^2 is the absolute value of x. So, we've got a definition of the square 80 00:05:59,343 --> 00:06:03,932 root function and we've seen that the square root of x^2 is not just x, 81 00:06:03,932 --> 00:06:05,900 it's the absolute value of x. Now, 82 00:06:05,900 --> 00:06:09,124 that doesn't actually address the original question, right? 83 00:06:09,124 --> 00:06:13,128 The original question is, what's the domain of this square root function? 84 00:06:13,128 --> 00:06:17,631 What sorts of numbers can I take root of? For instance, can I take the square root 85 00:06:17,631 --> 00:06:19,800 of a negative number? Let's see why not. 86 00:06:19,800 --> 00:06:23,266 Very concretely. Does it make sense, say, to talk about 87 00:06:23,266 --> 00:06:26,995 the square root of -16? Well, if it did that would be some 88 00:06:26,995 --> 00:06:29,873 number. So, I'll call that number k for crazy, 89 00:06:29,873 --> 00:06:33,078 alright? And what do I know about that number k? 90 00:06:33,078 --> 00:06:37,657 Well, k^22 would have to be -16. Remember, the definition of the square 91 00:06:37,657 --> 00:06:41,189 root function? It's a number that I square to get back 92 00:06:41,189 --> 00:06:45,245 the original number. So, if there were a square root of -16, 93 00:06:45,245 --> 00:06:50,341 when I square it, I get back -16. And imagining here that k is some real 94 00:06:50,341 --> 00:06:52,872 number. And that means there's three 95 00:06:52,872 --> 00:06:56,386 possibilities. Either k is positive, k is zero, or k is 96 00:06:56,386 --> 00:06:59,831 negative. If k is positive, then k squared would 97 00:06:59,831 --> 00:07:05,173 also be positive because a positive number times a positive number is still 98 00:07:05,173 --> 00:07:08,477 positive. But that can't be, because k squared is 99 00:07:08,477 --> 00:07:12,905 supposed to be -16. So, this first possibility doesn't 100 00:07:12,905 --> 00:07:16,209 happen. Now, if k were zero, then k squared would 101 00:07:16,209 --> 00:07:19,935 be zero, but k squared is supposed to be -16. 102 00:07:19,935 --> 00:07:22,230 So, k isn't zero. Is k negative? 103 00:07:22,230 --> 00:07:26,108 Well then, what's k squared? That would be a negative number times a 104 00:07:26,108 --> 00:07:29,223 negative number, and that would still be positive. 105 00:07:29,223 --> 00:07:33,546 And that can't be because k squared is supposed to be -16. 106 00:07:33,546 --> 00:07:38,128 So, this possibility also doesn't happen. So, all of our possibilities have been 107 00:07:38,128 --> 00:07:40,704 eliminated, alright? There can't be a real number k, 108 00:07:40,704 --> 00:07:44,888 which is the square root of -16. Because if k were positive, k squared 109 00:07:44,888 --> 00:07:47,571 would be positive but k squared has to be negative. 110 00:07:47,571 --> 00:07:51,541 k can't be zero because then k squared isn't negative and k can't be negative 111 00:07:51,541 --> 00:07:55,511 because then k squared is positive but k squared is supposed to be negative, 112 00:07:55,511 --> 00:07:59,589 alright? The upshot is that it just doesn't make any sense to talk about the 113 00:07:59,589 --> 00:08:04,753 square root of a negative number. In contrast, it does make sense to talk 114 00:08:04,753 --> 00:08:07,911 about the square root of zero, which is just zero, 115 00:08:07,911 --> 00:08:11,971 zero squared is zero. And it also makes sense to talk about the 116 00:08:11,971 --> 00:08:16,418 square root of positive numbers. So, to summarize the situation, we can 117 00:08:16,418 --> 00:08:21,574 say that the domain of the square root function is all the numbers between zero 118 00:08:21,574 --> 00:08:25,440 and infinity, including zero. So, I'm using the square bracket. 119 00:08:25,440 --> 00:08:29,630 But, of course, not including infinity because infinity is not a number. 120 00:08:29,630 --> 00:08:34,706 Sometimes, you're asked to calculate the domain of a function that's more 121 00:08:34,706 --> 00:08:37,912 complicated than, than just the square root of x. 122 00:08:37,912 --> 00:08:41,052 Let's see an example of that. So, let's try this. 123 00:08:41,052 --> 00:08:46,396 Let's try to find the domain of this function g, which is the square root of 124 00:08:46,396 --> 00:08:49,869 2x + 4. And remember, the domain consists of all 125 00:08:49,869 --> 00:08:52,674 the inputs for which the rule makes sense. 126 00:08:52,674 --> 00:08:56,950 So, I just have to think which x values makes sense for this rule? 127 00:08:56,950 --> 00:09:02,253 Well, in order to take the square root of 2x + 4, I'm going to need that 2x + four 128 00:09:02,253 --> 00:09:07,363 is not negative because I can't take the square root of a negative number so I 129 00:09:07,363 --> 00:09:12,407 need to guarantee that 2x + 4 is not negative, meaning greater than or equal 130 00:09:12,407 --> 00:09:15,512 to zero. Now, I can subtract four from both sides 131 00:09:15,512 --> 00:09:20,492 and I get that 2x is at least -4. Then, I can divide both sides by two. 132 00:09:20,492 --> 00:09:23,855 Two is positive, so it doesn't change the inequality. 133 00:09:23,855 --> 00:09:29,705 x is bigger than or equal to -2. So, as long as x is at least -2, then 2x 134 00:09:29,705 --> 00:09:36,286 + 4 is at least zero, which means it makes sense to take the square root. 135 00:09:36,286 --> 00:09:40,790 So, I can summarize the situation, the domain of g consists of all numbers 136 00:09:40,790 --> 00:09:44,987 greater than or equal to -2. This is our notation for that. 137 00:09:44,987 --> 00:09:50,348 I used a square bracket to include the -2 and the round bracket here on the 138 00:09:50,348 --> 00:09:54,740 infinity, because infinity is not number, it's not part of the domain. 139 00:09:54,740 --> 00:09:59,014 So, that example was a little bit harder. Let's do an even harder example where 140 00:09:59,014 --> 00:10:03,123 I've got multiple square roots, all right, the square root of something plus 141 00:10:03,123 --> 00:10:06,904 the square root of something. And let's figure out the domain of this 142 00:10:06,904 --> 00:10:09,315 function that has two separate square roots. 143 00:10:09,315 --> 00:10:14,082 This is the function T(x) equals the square root of 1 - x plus the square root 144 00:10:14,082 --> 00:10:16,720 of 1 + x. Now, in order for this rule to make 145 00:10:16,720 --> 00:10:20,729 sense, I have to be able to take this square root and also take this square 146 00:10:20,729 --> 00:10:23,187 root. In other words, in order to do this first 147 00:10:23,187 --> 00:10:27,250 square root, I'm going to need that 1 - x is bigger than or equal to zero, 148 00:10:27,250 --> 00:10:30,021 alright? I need the thing under the square root to 149 00:10:30,021 --> 00:10:32,516 be nonnegative in order to do a square root. 150 00:10:32,516 --> 00:10:37,117 In order to take this square root, I need 1 + x to be bigger than or equal to zero. 151 00:10:37,117 --> 00:10:41,496 And both of these things have to be true in order to take both of these square 152 00:10:41,496 --> 00:10:45,700 roots and then add them together. So, I'll put an and between them. 153 00:10:45,700 --> 00:10:50,109 Now, I go to x to both sides and this inequality and I get one is bigger than 154 00:10:50,109 --> 00:10:53,826 or equal to x. And I can subtract one from both sides of 155 00:10:53,826 --> 00:10:56,850 this and I'll get x is bigger or equal to -1. 156 00:10:56,850 --> 00:11:00,126 And again, both of these things have to happen, right? 157 00:11:00,126 --> 00:11:05,733 I need x less than one and x bigger than or equal to -1 in order to evaluate this 158 00:11:05,733 --> 00:11:08,697 function. Let me write this in in a more reasonable 159 00:11:08,697 --> 00:11:09,230 way, right? 160 00:11:09,230 --> 00:11:12,907 Instead of writing one bigger than or equal to x, I can write what I just said, 161 00:11:12,907 --> 00:11:16,011 x less than one. And here, I'll write, this is x bigger 162 00:11:16,011 --> 00:11:20,315 than or equal to -1. Now, I could write these inequalities as 163 00:11:20,315 --> 00:11:25,347 something about an interval. I could say that x is in the interval -1 164 00:11:25,347 --> 00:11:26,208 to 1, alright? 165 00:11:26,208 --> 00:11:32,035 To say that x is less than one and bigger than or equal to -1, exactly means that 166 00:11:32,035 --> 00:11:36,074 your inside this interval. And I'm using square brackets here, 167 00:11:36,074 --> 00:11:40,444 because I've got greater than or equal to, less than or equal to. 168 00:11:40,444 --> 00:11:45,344 And then, I can summarize the situation by writing the domain of T is this 169 00:11:45,344 --> 00:11:45,344 interval, alright? 170 00:11:45,344 --> 00:11:51,295 And this is describing the values of x for which this rule makes sense at the 171 00:11:51,295 --> 00:11:54,985 domain of the function T. Let's do one more example. 172 00:11:54,985 --> 00:11:59,545 some square root problem where I've also got an x squared term. 173 00:11:59,545 --> 00:12:02,873 Let's calculate the domain of this function C. 174 00:12:02,873 --> 00:12:09,959 C of x is the square root of 1 - x^2. So, the domain consists of all the inputs 175 00:12:09,959 --> 00:12:14,734 for which the rule makes sense. So, I'm looking for which values of x 176 00:12:14,734 --> 00:12:18,246 make the thing under the square root nonnegative. 177 00:12:18,246 --> 00:12:23,794 There's lots of different ways to think about which values of x make this true. 178 00:12:23,794 --> 00:12:31,659 one way is to factor 1 - x^2. So, I could factor 1 - x^2 as 1 + x * 1 - 179 00:12:31,659 --> 00:12:34,960 x, alright? That is equal to 1 - x^2. 180 00:12:34,960 --> 00:12:39,831 I'm looking for when that's nonnegative. This is a little bit easier to think 181 00:12:39,831 --> 00:12:44,029 about because now, I just got to figure out when these two terms have the same 182 00:12:44,029 --> 00:12:44,728 sign, alright? 183 00:12:44,728 --> 00:12:48,711 When they're both positive or they're both negative, then their product is 184 00:12:48,711 --> 00:12:51,510 bigger than or equal to zero. So, to think about that, 185 00:12:51,510 --> 00:12:59,223 I'll draw a number line. And I'll first think about when 1 + x is 186 00:12:59,223 --> 00:13:04,937 positive and negative. So, something special happens at -1, 187 00:13:04,937 --> 00:13:09,794 alright? When x is minus one, 1 + x is zero. 188 00:13:09,794 --> 00:13:14,841 When x is less than -1, 1 + x is negative. 189 00:13:14,841 --> 00:13:20,460 And when x is bigger than -1, 1 plus x is positive. 190 00:13:22,180 --> 00:13:30,785 Alright. Now, compare this with 1 - x, alright? 1 - x, something exciting 191 00:13:30,785 --> 00:13:38,060 happens at one, alright When X is less then one, 1 - x is positive. 192 00:13:39,320 --> 00:13:45,643 And when x is bigger than one, 1 - x is negative. 193 00:13:45,643 --> 00:13:52,648 Now, I'm not trying really to understand 1 + x or 1 - x, I'm trying to understand 194 00:13:52,648 --> 00:13:56,651 their product. So, when I multiply those two together, I 195 00:13:56,651 --> 00:14:03,580 get 1 - x^2 and I want to know, you know, when is that positive or or negative. 196 00:14:04,660 --> 00:14:08,503 Let me mark down the special points -1 and 1. 197 00:14:08,503 --> 00:14:15,337 And now, 1 - x^2 is the product of these so I can think about various values of x. 198 00:14:15,337 --> 00:14:22,099 So, when x is less than -1, then 1 + x is negative and 1 - x is positive, and a 199 00:14:22,099 --> 00:14:25,801 negative number times a positive number is negative. 200 00:14:25,801 --> 00:14:33,061 When x is between -1 and 1, then 1 + x is positive and 1 - x is also positive in 201 00:14:33,061 --> 00:14:35,980 that region, so their product is positive. 202 00:14:35,980 --> 00:14:41,803 And when x is bigger than one, 1 + x is positive and 1 - x is negative, so their 203 00:14:41,803 --> 00:14:45,621 product is negative. Now, this gets me most of the way there, 204 00:14:45,621 --> 00:14:48,792 alright? Because what I'm trying to understand is 205 00:14:48,792 --> 00:14:53,645 when this product is nonnegative and I can see that it's positive in this 206 00:14:53,645 --> 00:14:58,562 region, I could also think about what happens when I plug in -1 and 1. 207 00:14:58,562 --> 00:15:02,380 When I plug in -1, I get 1 - 1 which is zero. 208 00:15:02,380 --> 00:15:06,068 And when I plug in one, I get 1 - 1 which is zero. 209 00:15:06,068 --> 00:15:12,126 So, the function is, in fact, is zero in between here at -1 and 1. 210 00:15:12,126 --> 00:15:18,850 So, I'm just trying to figure out which values of x make 1 - x^2 nonnegative. 211 00:15:18,850 --> 00:15:25,197 Well, -1, 1, and anything in between. So, one way to summarize the situation is 212 00:15:25,197 --> 00:15:31,770 to say that the domain of my function C consists of all real numbers between -1 213 00:15:31,770 --> 00:15:37,581 and 1, including -1 and 1. So, I'm using the square brackets. 214 00:15:37,581 --> 00:15:46,863 As long as x is inside here, then 1 - x^2 is nonnegative. 215 00:15:46,863 --> 00:15:56,394 That means it makes sense to take the square root and that's the domain of C. 216 00:15:56,394 --> 00:15:56,394 [MUSIC]