1 00:00:00,000 --> 00:00:00,800 2 00:00:00,800 --> 00:00:03,950 So one of the the YouTube viewers asked me to 3 00:00:03,950 --> 00:00:04,610 do this problem. 4 00:00:04,610 --> 00:00:05,970 And it looked like a pretty good problem. 5 00:00:05,969 --> 00:00:08,349 So I thought I would record a quick video on it. 6 00:00:08,349 --> 00:00:15,135 So the problem's this: f of g of x -- hope I get this right 7 00:00:15,135 --> 00:00:23,789 -- is equal to 2 times the square root of x squared plus 8 00:00:23,789 --> 00:00:32,649 1 minus 1, all of that over the square root of x 9 00:00:32,649 --> 00:00:36,759 squared plus 1 plus 1. 10 00:00:36,759 --> 00:00:47,969 And f of x is equal to 2x minus 1 over x plus 1. 11 00:00:47,969 --> 00:00:50,890 And the question is, if we know that f of g of x is equal to 12 00:00:50,890 --> 00:00:55,560 this fairly complex-looking expression, and that f of x is 13 00:00:55,560 --> 00:00:59,980 equal to this, than what is g of x? 14 00:00:59,979 --> 00:01:02,529 Well, we could actually do this problem by kind 15 00:01:02,530 --> 00:01:04,849 of just looking at it. 16 00:01:04,849 --> 00:01:09,839 Because f of g of x, what we do is, we took -- everywhere we 17 00:01:09,840 --> 00:01:13,219 see an x, we replace it with g of x. 18 00:01:13,219 --> 00:01:15,750 So, f of g of x would look like this. 19 00:01:15,750 --> 00:01:17,230 And I'll do it in another color. 20 00:01:17,230 --> 00:01:20,400 So, let's see. f of g of x. 21 00:01:20,400 --> 00:01:22,690 And let's say we don't know what g of x is. 22 00:01:22,689 --> 00:01:23,359 Would be like this. 23 00:01:23,359 --> 00:01:28,500 Everywhere where we saw an x, we replace it with g of x. 24 00:01:28,500 --> 00:01:33,609 Because -- so it would be 2 times g of x. 25 00:01:33,609 --> 00:01:41,250 Minus 1 over g of x plus 1. 26 00:01:41,250 --> 00:01:44,250 And we also know that f of g of x is equal to this thing. 27 00:01:44,250 --> 00:01:44,870 Right? 28 00:01:44,870 --> 00:01:47,450 Here we just took the g of x and put it in of f of x, and 29 00:01:47,450 --> 00:01:49,320 we wrote the g of x in the expression. 30 00:01:49,319 --> 00:01:53,119 But we know that that is equal to this expression up here. 31 00:01:53,120 --> 00:01:56,910 So that is also equal to 2 times the square root of x 32 00:01:56,909 --> 00:02:01,869 squared plus 1, minus 1, all of that over x 33 00:02:01,870 --> 00:02:06,900 squared plus 1 plus 1. 34 00:02:06,900 --> 00:02:09,340 And I think, now, at this point, you can kind 35 00:02:09,340 --> 00:02:09,965 of pattern match. 36 00:02:09,965 --> 00:02:13,430 And you can see what g of x is. 37 00:02:13,430 --> 00:02:15,599 2 times something minus 1, right? 38 00:02:15,599 --> 00:02:18,409 2 times something minus 1. 39 00:02:18,409 --> 00:02:21,669 2 times something minus 1 in the numerator, and that's 40 00:02:21,669 --> 00:02:23,839 something plus 1 in the denominator. 41 00:02:23,840 --> 00:02:26,460 And that's something plus 1 in the denominator. 42 00:02:26,460 --> 00:02:31,689 So in either case, we have g of x is equal to the square 43 00:02:31,689 --> 00:02:34,129 root of x squared plus 1. 44 00:02:34,129 --> 00:02:37,240 And, actually, in the example he sent me, there was 45 00:02:37,240 --> 00:02:38,110 a bunch of choices. 46 00:02:38,110 --> 00:02:40,610 And if you were given the choices, then all you have to 47 00:02:40,610 --> 00:02:44,140 really do is take each of the choices for g of x and replace 48 00:02:44,139 --> 00:02:49,000 them in for x in this expression. 49 00:02:49,000 --> 00:02:52,199 And then see which of those choices for g of x ends 50 00:02:52,199 --> 00:02:55,709 up with this expression. 51 00:02:55,710 --> 00:02:56,450 Actually, we can do it. 52 00:02:56,449 --> 00:02:59,159 I mean, he gave the choice -- well, let me see. 53 00:02:59,159 --> 00:03:03,620 Let me delete some of this. 54 00:03:03,620 --> 00:03:11,230 55 00:03:11,229 --> 00:03:17,389 I want to do it, let's see if I can do it in 56 00:03:17,389 --> 00:03:18,269 -- oh, there we go. 57 00:03:18,270 --> 00:03:20,060 Better. 58 00:03:20,060 --> 00:03:21,659 All right. 59 00:03:21,659 --> 00:03:24,930 So, I wanted you to see that you could do the problem 60 00:03:24,930 --> 00:03:26,590 even if you weren't given any choices. 61 00:03:26,590 --> 00:03:29,110 But if you are given choices, the problem actually even 62 00:03:29,110 --> 00:03:32,120 becomes a little bit more straightforward. 63 00:03:32,120 --> 00:03:36,390 So the choices were -- actually, I erased some of it. 64 00:03:36,389 --> 00:03:41,729 It was given that f of x is 2x minus 1 over x plus 1. 65 00:03:41,729 --> 00:03:46,000 And then they said, which of the following is g of x. 66 00:03:46,000 --> 00:03:49,639 So, g of x is equal to. 67 00:03:49,639 --> 00:03:51,129 And they give these choices. 68 00:03:51,129 --> 00:03:59,849 a), square root of x, b), square root of x squared plus 69 00:03:59,849 --> 00:04:03,829 1, which we just figured out was actually the answer. 70 00:04:03,830 --> 00:04:10,740 c), x, d), x squared. 71 00:04:10,740 --> 00:04:15,360 And then e), x squared plus 1. 72 00:04:15,360 --> 00:04:17,710 And the way you would do this problem if, you, just by 73 00:04:17,709 --> 00:04:21,669 looking at it you didn't -- the method we just saw, saw that if 74 00:04:21,670 --> 00:04:25,540 you just replaced x with square root of x squared plus 1 75 00:04:25,540 --> 00:04:27,500 everywhere, then you'd get f of g of x. 76 00:04:27,500 --> 00:04:30,439 The other option is, you just take each of these and say well 77 00:04:30,439 --> 00:04:34,430 what is f of -- if you replaced x with this term everywhere, 78 00:04:34,430 --> 00:04:34,990 what do you get? 79 00:04:34,990 --> 00:04:38,230 Well, then you'd get 2 times square root of x minus 1 over 80 00:04:38,230 --> 00:04:40,050 the square root of x plus 1. 81 00:04:40,050 --> 00:04:43,090 And that's not what we have up here. 82 00:04:43,089 --> 00:04:46,669 Then if you took this and you replaced for x everywhere -- so 83 00:04:46,670 --> 00:04:49,689 if you took this expression, you replaced it for x 84 00:04:49,689 --> 00:04:53,899 everywhere, then you would get this expression. 85 00:04:53,899 --> 00:04:55,629 So you would know that this would be the answer. 86 00:04:55,629 --> 00:04:59,680 If you just replaced x with x, you would just get -- you would 87 00:04:59,680 --> 00:05:01,819 just get this over again, which does not equal this. 88 00:05:01,819 --> 00:05:03,259 So that doesn't work. 89 00:05:03,259 --> 00:05:09,810 If g of x was this, then f of g of x would be -- let's see, 90 00:05:09,810 --> 00:05:11,660 everywhere we see an x, you'd put an x squared, so it 91 00:05:11,660 --> 00:05:16,320 would be 2x squared minus 1 over x squared plus 1. 92 00:05:16,319 --> 00:05:17,980 Which does not equal this. 93 00:05:17,980 --> 00:05:22,530 And similarly, if g of x was this term right here, then f of 94 00:05:22,529 --> 00:05:32,599 g of x would be would be 2 times x squared plus 1 minus 1 95 00:05:32,600 --> 00:05:37,070 over x squared plus 1 plus 1. 96 00:05:37,069 --> 00:05:39,149 Which you can simplify a bit, but that still 97 00:05:39,149 --> 00:05:40,599 doesn't equal this. 98 00:05:40,600 --> 00:05:42,890 And so this, once again, if you're given the choices, 99 00:05:42,889 --> 00:05:43,779 you just try this out. 100 00:05:43,779 --> 00:05:46,199 And if you replace this expression everywhere, 101 00:05:46,199 --> 00:05:47,920 where you see an x, right? 102 00:05:47,920 --> 00:05:51,840 Everywhere where you see an x, and you replace it this 103 00:05:51,839 --> 00:05:55,199 expression, you get these and that's what the question, 104 00:05:55,199 --> 00:05:57,000 essentially, was asking us. 105 00:05:57,000 --> 00:06:00,000 So, hopefully I didn't confuse everyone too much. 106 00:06:00,000 --> 00:06:03,730 And hopefully this was helpful for the viewer 107 00:06:03,730 --> 00:06:05,080 who asked me to do this. 108 00:06:05,079 --> 00:06:05,939 I'll see you all later. 109 00:06:05,939 --> 00:06:07,240 Bye. 110 00:06:07,240 --> 00:06:07,281