WEBVTT
1
00:00:00.000 --> 00:00:07.000
On this video we do the following five composite functions.
2
00:00:07.000 --> 00:00:16.000
This is part seven of functions and we're going to be doing some more examples with composite functions.
3
00:00:16.000 --> 00:00:23.000
Remember the definition, the notation, by write the parentheses f of g of x. Remember this little circle here.
4
00:00:23.000 --> 00:00:30.000
That's a composite function. That just means f of g of x.
5
00:00:30.000 --> 00:00:30.000
Let's try some examples.
6
00:00:30.000 --> 00:00:43.000
Let's say we've got these three functions, f of x, equals x squared minus 3x plus 5, g of x equals negative 4x plus 1, and h of x equals square root of x.
7
00:00:43.000 --> 00:00:50.000
Let's start out with finding f of g of negative 2.
8
00:00:50.000 --> 00:01:00.000
The first thing we need to do is write down what that means. That means f of g of negative 2.
9
00:01:00.000 --> 00:01:13.000
I need to figure out what g of negative 2 is. I can go over on the side. You don't have to do this, but I can figure out what g of negative 2 is by plugging in negative 2 into the function for g.
10
00:01:13.000 --> 00:01:20.000
Remember, g of x was negative 4x plus 1.
11
00:01:20.000 --> 00:01:29.000
G of negative 2 is negative 4 times negative 2 plus 1. That gives you 8 plus 1 or 9.
12
00:01:29.000 --> 00:01:36.000
When you plug in negative 2 for x into the function for g, we're going to get 9.
13
00:01:36.000 --> 00:01:46.000
Now we go back to finish this problem. Now that we know of g of negative 2 is 9, I really need to figure out what f of 9 equals.
14
00:01:46.000 --> 00:01:56.000
Now, let's look back up to the top here. f of x is x squared minus 3x plus 5. We're going to plug in 9 for x.
15
00:01:56.000 --> 00:02:09.000
That gives you 9 squared minus 3 times 9 plus 5. We have 81 minus 27 plus 5.
16
00:02:09.000 --> 00:02:25.000
You can go left to right, 81 minus 27 is 54, add 5, and the answer is 59.
17
00:02:25.000 --> 00:02:32.000
That's how we would do f of g of negative 2.
18
00:02:32.000 --> 00:02:38.000
Here's another example. Same three functions, but I'm asking for h of g of negative 2.
19
00:02:38.000 --> 00:02:45.000
See if you can do this on your own by putting the video on pause and then coming back to it.
20
00:02:45.000 --> 00:02:55.000
This means h of g of negative 2. I need to figure out what g of negative 2 is.
21
00:02:55.000 --> 00:03:06.000
We're going to look at the g of x is negative 4x plus 1. g of negative 2 means I'm plugging in negative 2 for x.
22
00:03:06.000 --> 00:03:13.000
Notice this is exactly what we did. A second ago I've got the equal sign. This is equal to 8 plus 1 or 9.
23
00:03:13.000 --> 00:03:20.000
From the previous problem we also got that the g of negative 2 is 9. I'm just doing the same steps here.
24
00:03:20.000 --> 00:03:33.000
Then we want h of 9. g of negative 2 is 9. I'm plugging that in. 9 for g of negative 2.
25
00:03:33.000 --> 00:03:42.000
Now we have to look at what's h of 9. We have to go back up to the function for h, which is the square root of x.
26
00:03:42.000 --> 00:03:50.000
h of 9 would be the square root of 9. The square root of 9 is 3.
27
00:03:50.000 --> 00:04:01.000
This is not number 1. This is number 2. For h of g of negative 2 is 3.
28
00:04:01.000 --> 00:04:07.000
Let's use these same three functions. And compute g of f of 4.
29
00:04:07.000 --> 00:04:12.000
Again try this on your own by putting the video on pause.
30
00:04:12.000 --> 00:04:19.000
Remember this means g of f of 4. We need to figure out f of 4.
31
00:04:19.000 --> 00:04:31.000
Now remember, since f of x is x squared minus 3x plus 5, then f of 4 would be 4 squared minus 3 times 4 plus 5.
32
00:04:31.000 --> 00:04:43.000
We're just plugging in 4 for x into the function for f. So 16 minus 12 plus 5 is, well 16 minus 12 is 4 plus 5 more will give it 9.
33
00:04:43.000 --> 00:04:50.000
So I'm going to plug in 9 for f of 4. Come back over here. You don't have to do this on the side.
34
00:04:50.000 --> 00:04:58.000
You could write all this within that parenthesis, but it seems like it helps people to do the side work.
35
00:04:58.000 --> 00:05:06.000
So we have g of, now I know f of 4 is 9. Right now I have to look at the function for g.
36
00:05:06.000 --> 00:05:13.000
g of x is negative 4x plus 1. So I'm going to plug in 9 for x.
37
00:05:13.000 --> 00:05:26.000
So I have negative 4 times 9 plus 1. Negative 36 plus 1 is negative 35.
38
00:05:26.000 --> 00:05:33.000
So hopefully you're getting the hang of it. Let's do some more.
39
00:05:33.000 --> 00:05:40.000
All right, here's the next one. Same three functions. How do you do h of f of negative 5?
40
00:05:40.000 --> 00:05:44.000
Go ahead and put the video on pause and try this on your own first.
41
00:05:44.000 --> 00:05:49.000
Okay, so what's this mean? It means h of f of negative 5.
42
00:05:49.000 --> 00:05:53.000
So I'm going to go over to the side again to figure out what f of negative 5 is.
43
00:05:53.000 --> 00:05:59.000
So remember f of x equals x squared minus 3x plus 5.
44
00:05:59.000 --> 00:06:04.000
So we want h of negative 5. So we're going to have to put a negative 5 for x.
45
00:06:04.000 --> 00:06:09.000
Now remember we're doing x squared. So we have to do negative 5 times negative 5.
46
00:06:09.000 --> 00:06:19.000
So I'm going to write that. Negative 5 times negative 5. That's going to be x squared minus 3 times negative 5 plus a 5.
47
00:06:19.000 --> 00:06:30.000
So what's that give us? 25 plus 15 plus 5, which is 45.
48
00:06:30.000 --> 00:06:37.000
So in place of f of negative 5 here, we're going to plug in 45.
49
00:06:37.000 --> 00:06:41.000
So we get h of 45.
50
00:06:41.000 --> 00:06:49.000
Now we look at the h function. h of x is square root of x. So we want the square root of 45.
51
00:06:49.000 --> 00:06:57.000
Is that simplified? Nope. Because it has a factor of 9 and 45, which is a perfect square.
52
00:06:57.000 --> 00:07:02.000
So I think of this as the square root of 9 times 5.
53
00:07:02.000 --> 00:07:06.000
And I could take a 3x. That's the square root of 9.
54
00:07:06.000 --> 00:07:17.000
So the final answer is 3 square root of 5.
55
00:07:17.000 --> 00:07:24.000
All right. What if we want to use the same three functions and let's find f of g of x?
56
00:07:24.000 --> 00:07:27.000
Okay. So we're not plugging in a number at this time.
57
00:07:27.000 --> 00:07:37.000
So we have f of g of x. And now we know what g of x is. We just look up here.
58
00:07:37.000 --> 00:07:42.000
And g of x is negative 4x plus 1. So that's what we replace g of x with.
59
00:07:42.000 --> 00:07:46.000
So we're just going to have to still deal with the variables.
60
00:07:46.000 --> 00:07:50.000
And then we want f of negative 4x plus 1.
61
00:07:50.000 --> 00:07:55.000
So we have to go back up to the function f up here.
62
00:07:55.000 --> 00:08:00.000
And we're going to plug in negative 4x plus 1 everywhere. There's an x.
63
00:08:00.000 --> 00:08:13.000
So we have negative 4x plus 1 squared minus 3 times negative 4x plus 1 plus 5.
64
00:08:13.000 --> 00:08:16.000
And then we have to be careful when you're scoring this binomial.
65
00:08:16.000 --> 00:08:23.000
You forget your middle term. That'll give you a 16x squared minus 8x plus 1.
66
00:08:23.000 --> 00:08:28.000
Remember, you could do that as negative 4x plus 1 times negative 4x plus 1.
67
00:08:28.000 --> 00:08:34.000
Do the foil method get all four terms and then you'll get that correct middle term of negative 8x.
68
00:08:34.000 --> 00:08:37.000
All right. We've got the distributive property here.
69
00:08:37.000 --> 00:08:43.000
Negative 3 times negative 4 is plus 12x. And the negative 3 times 1 is negative 1.
70
00:08:43.000 --> 00:08:49.000
I'm sorry, negative 3. And we have plus 5 hanging out there. Almost done.
71
00:08:49.000 --> 00:08:54.000
All we have to do is add like terms. So we have 16x squared.
72
00:08:54.000 --> 00:08:58.000
Let's say we've got a negative 8 plus 12. That's the x terms.
73
00:08:58.000 --> 00:09:01.000
So that's going to be plus 4x.
74
00:09:01.000 --> 00:09:07.000
And let's say we have 1 minus 3 and a plus 5. That's plus 3.
75
00:09:07.000 --> 00:09:19.000
And that is your answer.
76
00:09:19.000 --> 00:09:21.000
Now here's something interesting.
77
00:09:21.000 --> 00:09:26.000
The first problem we did was f of g of negative 2. And we got 59.
78
00:09:26.000 --> 00:09:34.000
These were the steps we used. We just figured out f of g of x and got 16x squared plus 4x plus 3.
79
00:09:34.000 --> 00:09:42.000
So if you first figure out f of g of x, you could then just plug in the negative 2 into this one function.
80
00:09:42.000 --> 00:09:47.000
So just so you can see that that works. That will give you 16 times x squared.
81
00:09:47.000 --> 00:09:54.000
That's a negative 2 squared plus 4 times a negative 2 plus 3.
82
00:09:54.000 --> 00:09:59.000
Now the quantity negative 2 squared, negative 2 times negative 2 is 4.
83
00:09:59.000 --> 00:10:04.000
And then we have plus and negative 8 or minus 8 and plus 3.
84
00:10:04.000 --> 00:10:12.000
So that gives me 64 minus 8 plus 3, which is also 59.
85
00:10:12.000 --> 00:10:18.000
You should get the same answer if you wanted to first compute f of g of x.
86
00:10:18.000 --> 00:10:23.000
Now that would be convenient if you were doing a whole bunch of problems like f of g of 3,
87
00:10:23.000 --> 00:10:27.000
f of g of 9, f of g of a bunch of things. But if it's just for negative 2,
88
00:10:27.000 --> 00:10:33.000
it wasn't so bad. You just do g of negative 2 and then plug that in and we'll get 59.
89
00:10:33.000 --> 00:10:37.000
But this is just something to pay attention to that.
90
00:10:37.000 --> 00:11:06.000
This also gives you the correct answer.