10 - What are Composite Functions? (Part 1) - Evaluating Composition of Functions & Examples - By Math and Science
Transcript
00:00 | Hello . Welcome back . The title of this lesson | |
00:03 | is called composite functions . This is part one of | |
00:06 | two . Now I'm excited to teach this because I | |
00:08 | get lots and lots of correspondence with students . They | |
00:11 | get really confused with composite functions . Let me break | |
00:14 | it down in the beginning and tell you that the | |
00:15 | concept is very very simple to understand . We have | |
00:18 | to go through some background material before you kind of | |
00:20 | kind of clicks with you . Remember a function in | |
00:23 | general is like a black box . It's like a | |
00:26 | mathematical box . Inside that box is a mathematical calculation | |
00:30 | called a function to the input of the box are | |
00:33 | input values . We call those X values . Those | |
00:36 | X values are fed one at a time into this | |
00:38 | box . Inside the box the calculation happens you know | |
00:42 | ffx and the calculation you know happens according to whatever | |
00:45 | the function is . And then on the output you | |
00:47 | get the corresponding output which is what we call F | |
00:50 | of X . And we can plot X versus F | |
00:52 | of X . And that's what we get . We | |
00:53 | get a graph of all the input functions in showing | |
00:56 | what all the output functions are . A composite function | |
00:59 | at the biggest overview is basically if you take two | |
01:03 | separate functions , totally different functions , we call one | |
01:06 | of them F of X and we call one of | |
01:08 | them G of X . We have to give them | |
01:10 | different names because if we call them both , ffx | |
01:12 | will get confused . There are two different functions . | |
01:15 | And then what we're gonna do is we're going to | |
01:16 | change those functions together . We're going to send the | |
01:19 | input into one function . We're gonna get the output | |
01:21 | , we're gonna take that output , we're gonna send | |
01:23 | it right back into another function over here and we're | |
01:26 | gonna get its output . So together those two functions | |
01:29 | operate as a team input comes in into one function | |
01:32 | . You get an output and intermediate output . But | |
01:35 | that output goes right into the next function and we | |
01:37 | get the final output composite function means we take two | |
01:40 | things and make a composite calculation which is kind of | |
01:43 | a combination of the two , this composite function thing | |
01:46 | we learned in algebra and pre calculus but we use | |
01:49 | it extensively and calculus because very quickly the problems get | |
01:53 | much more complicated and you have to understand what a | |
01:55 | composite function is to do almost anything in calculus . | |
01:58 | So let's take it with a very very simple example | |
02:02 | . Let's say we have a function let's call it | |
02:05 | G . Of X . Because we have to have | |
02:07 | different names F of X and G . Fx . | |
02:08 | We're gonna say this function is equal to two times | |
02:10 | X . And we're gonna say we have another function | |
02:13 | called F of X . And this function is X | |
02:16 | squared . So the job of this function is to | |
02:19 | take whatever I stick into this function and just square | |
02:22 | it . All it does is it squares the input | |
02:24 | . If I put a two in there I get | |
02:25 | a four . If I put a three in there | |
02:27 | I get a nine . If I put a four | |
02:28 | in there I get a 16 and so on . | |
02:30 | This function in the job of it is just to | |
02:32 | double the input . If I put a two in | |
02:34 | I get a four . If I put a three | |
02:36 | in I get a six . If I put a | |
02:38 | four in I get an eight because whatever I put | |
02:41 | into that g function I just double it right now | |
02:44 | . What we're gonna do is link these functions together | |
02:46 | , we're gonna send an input into here , get | |
02:48 | an output and we're gonna send it right back into | |
02:50 | the other function and get the final output . And | |
02:53 | that thing that process is called a composite function . | |
02:56 | So graphically what we're gonna do is we're gonna take | |
02:59 | X values and we're gonna stick them in not just | |
03:02 | into one function but into two functions . The first | |
03:05 | function that we're going to send them into is the | |
03:08 | G of X functions . We're gonna get some answer | |
03:11 | right . Whatever the answer is , we're gonna get | |
03:13 | it but we're gonna send that answer directly into another | |
03:16 | box . And in this box contains the F of | |
03:19 | X function , right ? And then this is the | |
03:23 | part that usually confuses students but should shouldn't confusion . | |
03:26 | Now the output of this thing is written like this | |
03:30 | F of G of X . You see the way | |
03:36 | you need to read this is , you know , | |
03:38 | order of operations . Remember order of operations When you | |
03:41 | have nested parentheses . What do you do you look | |
03:44 | at the inside parentheses first . If you have some | |
03:47 | addition going on and the innermost parentheses , you have | |
03:49 | to do it first . Then you go a little | |
03:52 | bit beyond and a little bit beyond expanding and going | |
03:54 | through the princess from inside to outside . Same thing | |
03:57 | here you have to put the uh value into the | |
04:01 | innermost function first evaluate the answer . But whatever you | |
04:05 | get out of that is what is sent into the | |
04:07 | F function because remember F of X . Whatever is | |
04:09 | in here gets passed to the F function . So | |
04:12 | first you evaluate G . You get an answer and | |
04:16 | you stick it into the F function , go from | |
04:17 | inside to outside graphically . You stick on number in | |
04:20 | calculate G of X . That number gets fed into | |
04:23 | ffx . But this is how you write it . | |
04:25 | It's called a composite function . We're not gonna be | |
04:27 | drawing pictures , you're gonna be writing this . You | |
04:29 | need to know you work inside to outside . All | |
04:33 | right now these are called I'll just write it down | |
04:36 | composite functions because there are composite of two other functions | |
04:45 | . But I think I have some better terminology for | |
04:48 | you . Uh actually need to learn how to spell | |
04:51 | composite . There's an E . Right there . Um | |
04:54 | I like to say that they are chained functions and | |
05:05 | you can see why because you do one then you | |
05:06 | do the other . And by the way you can | |
05:08 | have a composite function with more than two functions . | |
05:10 | You can take an input going into one function and | |
05:13 | feeds into another function feeds into a third function . | |
05:15 | Maybe it even feeds into a force function and you | |
05:17 | get the final answer . Those are kind of chained | |
05:20 | together . So I like to actually call them chained | |
05:22 | functions . But even a better word than change functions | |
05:26 | would be this nested functions if you like this picture | |
05:35 | better . It looks more like a chain . But | |
05:37 | if you like this picture better , it looks more | |
05:39 | like a nested nested , like in computer program nest | |
05:42 | nested means one thing inside of another , inside of | |
05:44 | another , inside of another . They're nested inside . | |
05:46 | But you always have to go into the innermost thing | |
05:49 | first . Just like order of operations and work your | |
05:51 | way out . All right , so , we have | |
05:53 | a concrete example . We have these two functions . | |
05:56 | Uh And what I'm gonna do is write them down | |
05:58 | again because I've used so much board space . I | |
06:00 | want to make sure and have it very uh clear | |
06:03 | here . G of X is two . X . | |
06:07 | Eh Fedex is X squared . So now let's do | |
06:11 | some real calculations and show how easy this really is | |
06:13 | . If you take an input value of X is | |
06:15 | equal to two , I can pick any number at | |
06:17 | what I'm gonna stick a number of X is equal | |
06:19 | to two and I'm gonna stick it into G of | |
06:21 | X . So I have to evaluate G at X | |
06:24 | is equal to two , then I'm gonna stick it | |
06:26 | in here and then two times two is gonna give | |
06:28 | me four , but then I'm gonna take the output | |
06:30 | of this guy and I'm gonna feed it directly into | |
06:33 | uh the function F . Of X . But really | |
06:37 | what you're doing is you're saying F of G of | |
06:39 | X because what you're doing is the argument , what's | |
06:42 | going into the f function is whatever you calculated prior | |
06:45 | to that . So it's going to be F of | |
06:48 | G of two because that's the number I use . | |
06:50 | So I got a four here , so I'm evaluating | |
06:53 | F of G F two , which means I'm evaluating | |
06:55 | the function at the number four , I'm taking this | |
06:58 | and I'm sticking it right into this function And that | |
07:00 | is equal to 16 . Why ? Because when I | |
07:02 | take this and stick it in here , four times | |
07:04 | four is 16 . All right , let's do another | |
07:08 | one . What if I say X is equal to | |
07:10 | three ? Well , I have to go into G | |
07:14 | first evaluated at three . I'm gonna double it . | |
07:17 | So three plus three or three times two is six | |
07:20 | . I'm gonna take this and then I have to | |
07:21 | feed it into the next function . So F . | |
07:23 | Of G of three . Right ? Because I take | |
07:27 | whatever I got in the output and I stick it | |
07:29 | in here means it's F of whatever I calculated before | |
07:34 | . And then six goes right into here . Six | |
07:36 | times six is 36 . So , you would say | |
07:38 | F of G of three is 36 . F of | |
07:40 | G . Of two is 16 . Now . We've | |
07:44 | done it with numbers . We can stick numbers in | |
07:47 | here and get numbers all day long . But what | |
07:49 | if we keep it a little bit more general ? | |
07:50 | What if we just feed a general value of X | |
07:53 | . N . We're not gonna say X . Is | |
07:54 | equal to one or X is equal to two or | |
07:56 | X is equal to negative three . We're just gonna | |
07:58 | keep it general because remember , a function can take | |
08:00 | any value of X . N . That's in its | |
08:02 | domain . So let's just keep it general say let's | |
08:05 | feed the value of X . M . We're gonna | |
08:07 | again go and feed into the G . Function . | |
08:10 | But in this case we're putting a generic thing and | |
08:12 | so we're gonna make it G . Of X . | |
08:13 | Because that's what we're sticking in , right ? And | |
08:16 | what is that going to be ? Well , we | |
08:17 | stick this in here and we're gonna get to X | |
08:20 | out . Whatever goes in gets multiplied by two . | |
08:22 | So the answer to G of X two X , | |
08:25 | we're gonna take this and we're gonna feed it in | |
08:27 | here , F of G of X Is going to | |
08:32 | be f of whatever I got here , two x | |
08:37 | . And whenever I put it in here , what | |
08:39 | am I going to get ? I'm gonna get to | |
08:41 | X quantity squared because whatever I put in to the | |
08:44 | f function is what is square , This whole thing | |
08:47 | is here . So the whole thing is squared . | |
08:49 | So , kind of a summary , you can see | |
08:52 | it very clearly with numbers what we're doing by changing | |
08:54 | these functions together , students sometimes get a little bit | |
08:57 | more confused whenever we just keep it as variables . | |
09:00 | But what we're basically saying is that uh in conclusion | |
09:04 | , I would say so dot dot dot right F | |
09:07 | of G of x is called a composite function of | |
09:11 | the g function evaluated first , then the f function | |
09:14 | . And what we get from that by doing it | |
09:16 | and running it through there is we get to x | |
09:18 | quantity squared or you can write it and say that | |
09:21 | F of G of x is equal to , we | |
09:25 | have something squared . We can apply the exponent here | |
09:28 | and get four x squared . We just apply the | |
09:30 | exponents of both things for x squared . So this | |
09:33 | is called a composite function . Notice that what we're | |
09:36 | saying is that we have two functions . One of | |
09:38 | them is G fx . One of them is ffx | |
09:39 | when we combine them together in this way , so | |
09:42 | that we first evaluate the g functions , then evaluate | |
09:45 | the F function . The resulting function is not the | |
09:49 | g function , and the resulting function is not the | |
09:52 | X function . It's kind of like a baby function | |
09:54 | that's like a daughter or a son product of the | |
09:57 | original two functions , right ? Four X squared is | |
10:00 | kind of like this and kind of like this , | |
10:03 | but it really has characteristics of both . And so | |
10:06 | this is called a composite function . So if something | |
10:09 | on your test said , hey , you have a | |
10:10 | G function and an F function , tell me what | |
10:12 | F of G of X is , all you have | |
10:13 | to do is take this guy sticking into here . | |
10:16 | And this is the general thing that you would get | |
10:18 | back . You can see that it's nested in the | |
10:20 | way that we write it here you go inside to | |
10:22 | outside . Or you can think of , it doesn't | |
10:25 | change however you like to think about it . Um | |
10:29 | Now what we need to do is solve some problems | |
10:31 | because everybody can kind of get the general idea of | |
10:34 | it . But until we do a few problems , | |
10:36 | it's gonna be a little bit fuzzy . So now | |
10:38 | let's turn our attention and let's redefine the two functions | |
10:41 | we have . So no longer are we talking about | |
10:43 | these two functions ? Now we're gonna change the functions | |
10:45 | and make them a little bit different than these . | |
10:47 | So we can get a little practice . So , | |
10:50 | I give this to you . Given F of X | |
10:55 | Is three X -5 . G of X . Is | |
11:00 | the square root of X . All right , let's | |
11:04 | take a look . Let's do a part a here | |
11:07 | . What would f of G of four be equal | |
11:12 | to remember ? You can think of it as nested | |
11:14 | or change however you want to think of it . | |
11:15 | But ultimately , you got to go on the inside | |
11:17 | first and do this first . So , this is | |
11:19 | what I recommend that you do look at the innermost | |
11:22 | princes and say , well , first I have to | |
11:23 | figure out what G . F four is equal to | |
11:25 | . I have a definition of G right here , | |
11:28 | so it's going to be the square root of whatever | |
11:29 | is here . Plugging into this location . So G | |
11:33 | of four Is going to be equal to two . | |
11:36 | That's what that's equal to . Yeah . And then | |
11:39 | directly kind of like the next part of the process | |
11:43 | is you're going to go and say , well , | |
11:45 | then I have to take this and plug it into | |
11:47 | the F function . So F of G of four | |
11:51 | is F of whatever I calculated in the previous answer | |
11:55 | , which is this thing , which is to So | |
11:58 | I now have to take that to and plug it | |
12:00 | into the F function . Which means it's going to | |
12:03 | be three times two minus five because I'm just sticking | |
12:07 | this value exactly where the X . Is , so | |
12:10 | three times two is six minus five which is equal | |
12:13 | to one . So what you have figured out is | |
12:16 | that F . Of G . Of four is equal | |
12:21 | to one . This is the final answer . So | |
12:24 | you're asked to calculate what this composite function is , | |
12:26 | literally all you have to do is look at the | |
12:28 | innermost thing , calculate that first , take the answer | |
12:31 | , run it through the other function , which is | |
12:33 | the slightly most outermost function and calculate the answer . | |
12:36 | You want to believe the number of students that I | |
12:38 | get with this confusion on this topic because it looks | |
12:42 | really complicated double parentheses , different functions . This so | |
12:45 | look so hard . But if you just look in | |
12:47 | the inside work your way out , it's all going | |
12:49 | to be the same thing . Mhm . All right | |
12:51 | , so now what we wanna do is uh calculate | |
12:55 | , I'm gonna do the exact same thing . I'm | |
12:57 | gonna say , I'm gonna rewrite the function . So | |
12:58 | ffx Is three X -5 G . of X . | |
13:04 | Is the square root of X . So I'm gonna | |
13:06 | we're gonna do a function . That's a calculation is | |
13:08 | very similar right here . Instead of F of G | |
13:11 | . F four , Let's do G of F of | |
13:15 | four . Now , a lot of students get confused | |
13:18 | because you look at these two things F of G | |
13:21 | four G . Of F . Of four and you | |
13:23 | think , oh well it's gonna be the same exact | |
13:25 | thing is going to be , what did I get | |
13:26 | here ? One is gonna be one . Right , | |
13:28 | and a lot of times , I think students think | |
13:30 | this , because you think about multiplication , three times | |
13:33 | two is 62 times three is six . Okay , | |
13:36 | same thing , two plus three is five , three | |
13:39 | plus two is five . Same thing . So it | |
13:41 | seems that in math things often happen in reverse order | |
13:44 | . No problem . But for composite functions it's not | |
13:46 | the case . You cannot just reverse the order of | |
13:49 | things and get the same , calculate the same answer | |
13:51 | . Let's see why we have to look at the | |
13:54 | innermost parentheses first . So we have to calculate F | |
13:57 | of four . First F is this function ? So | |
14:01 | it's going to be three times the four minus the | |
14:03 | five . Right ? So f of four Is going | |
14:09 | to be equal , this is 12 -5 , so | |
14:12 | f of four is going to be equal to seven | |
14:16 | . Okay . And then whatever I get at the | |
14:18 | output of this function , I got to run it | |
14:20 | through the outermost function which is G . So then | |
14:22 | you say G of F of four is going to | |
14:27 | be equal to whatever I got for this , Which | |
14:30 | means GF seven Stick it in there . Right ? | |
14:33 | And what is that ? I stick a seven into | |
14:35 | the G function . That's the square root of seven | |
14:38 | . And that's the answer squared of seven . So | |
14:40 | you get g . of f . of four Is | |
14:44 | equal to the square root of seven . And I | |
14:45 | know that you can see that this answer is completely | |
14:47 | and totally different than this one . So in general | |
14:51 | in general F of G of X is not equal | |
14:58 | to G of F . Of X . You see | |
15:01 | how this thing looks . So complicated . Even when | |
15:03 | I write it down like oh my gosh , it's | |
15:04 | so complicated . But now that we've done a couple | |
15:07 | and you can see what I'm talking about , all | |
15:09 | it's saying is that if I make a composite function | |
15:11 | where the innermost is G . And the outermost is | |
15:13 | death . And make another composite function where the innermost | |
15:16 | is F . Done first . And then going into | |
15:18 | G . And you do those calculations , you're not | |
15:20 | gonna get the same thing , I'm gonna write down | |
15:22 | here in general . It turns out there's actually a | |
15:29 | very important exception to this . And we're gonna get | |
15:32 | to that very soon . It's called an inverse function | |
15:35 | . When I craft a very specially created function called | |
15:39 | an inverse function . Then it turns out that these | |
15:41 | can actually be equal . But in general , for | |
15:43 | two random functions , I just pull off the shelf | |
15:46 | , I just pull this one from over here . | |
15:47 | This one over here , randomly . Then the composite | |
15:50 | functions when you flip them around and are not gonna | |
15:52 | be the same thing . All right . So we've | |
15:55 | done a couple of these with numbers . Now I | |
15:57 | want to do just a little bit to wrap it | |
15:59 | up with variables . So let's rewrite these functions again | |
16:02 | , The ffx function was three x minus five . | |
16:06 | The g fx function was the square root of X | |
16:09 | . Now , what I want to calculate next , | |
16:11 | uh for part C you could say is F of | |
16:16 | G of X . You might say , didn't we | |
16:18 | just do that ? Well , no , we calculated | |
16:20 | F of G of the number four , then we | |
16:22 | calculated G of f of the number four . Now | |
16:24 | we're changing it , so we're not just putting a | |
16:26 | number and we're just gonna leave it general and say | |
16:28 | it could be anything , we're gonna call it X | |
16:30 | . So how do you do that ? Well , | |
16:32 | you first have to say , well I put the | |
16:34 | value into G and I'm gonna get whatever I get | |
16:37 | out of G , what am I going to get | |
16:38 | out of G ? I'm gonna get the square root | |
16:40 | of X . Whatever I put in here , I'm | |
16:42 | gonna get the squared of X . Um So what | |
16:44 | you're going to have here is f . Of the | |
16:47 | square root of X . Because that's what G . | |
16:49 | Of X . Is equal to you . Take this | |
16:51 | and you have to pass that into the F . | |
16:54 | Function . So then you have to put it into | |
16:57 | here three times the square root of X minus five | |
17:01 | . I take whatever is in here and I stick | |
17:03 | it into the F . Function . Then when I | |
17:04 | get is F . Of G . Of X is | |
17:08 | equal to three times the square root of X minus | |
17:11 | five . This is what you would circle on your | |
17:13 | paper , F . of G . of X is | |
17:16 | three times a squared of X -5 . Now the | |
17:21 | next part of the problem , the d . Part | |
17:24 | is we want to flip this order around G . | |
17:26 | Of F . Of X . Here we did G | |
17:29 | first then plug it into X . Here , we're | |
17:32 | gonna do F first , then plug it into G | |
17:34 | . What do you think we're gonna get ? You | |
17:35 | think we're gonna get the same exact thing ? No | |
17:37 | , you're not because in general when you do the | |
17:39 | order of the composite backwards , it's gonna be a | |
17:42 | different answer . And so that's what we're gonna see | |
17:44 | here . If we look at the innermost function , | |
17:47 | it's just ffx . We stick an F . We're | |
17:48 | gonna stick an X . In , we get ffx | |
17:51 | out . So what we have to do is pass | |
17:53 | whatever this is into the G function , It's gonna | |
17:56 | be g . of this three X -5 because the | |
18:01 | FFX function is equal to this . Then we have | |
18:05 | to take whatever is here and stick it into the | |
18:06 | G function , which is just the square root of | |
18:09 | that . So what we're going to get is G | |
18:12 | . Of F . Of X is equal to the | |
18:16 | square root of three x minus five . You see | |
18:19 | if we take this entire thing and that goes into | |
18:21 | the g function , which means it's a square root | |
18:24 | of that whole thing . So this is the answer | |
18:26 | . So F of G of x is three times | |
18:28 | the square root of x minus five , And G | |
18:30 | of F of X is three times X -5 . | |
18:33 | And then the radical around the whole thing . Now | |
18:35 | , I know that you think this kind of looks | |
18:36 | sort of like this , but because the radical is | |
18:38 | in a different location , you agree with me that | |
18:40 | these are gonna give different answers and that's exactly why | |
18:43 | we got the different answers when we just stuck to | |
18:45 | numbers there . So here we introduce the very important | |
18:49 | concept of a composite function . I know it's hard | |
18:51 | to see why we care about composite functions in the | |
18:53 | beginning . We have to crawl before we can walk | |
18:56 | . The truth is they're important because they feed into | |
18:59 | what an inverse function is very soon . And inverse | |
19:02 | functions feed into law algorithms and algorithms feed into other | |
19:05 | things . So we're building , we're kind of like | |
19:06 | building the foundation . We're down here at the bottom | |
19:08 | , building the foundation , understand composite functions and make | |
19:12 | sure you can solve all of these problems . Follow | |
19:14 | me on to the next lesson . We're going to | |
19:15 | get some more practice with composite functions before moving on | |
19:18 | to the concept of an inverse function . And as | |
19:21 | I said before , in general , when you flip | |
19:23 | the order of the composite here , F of G | |
19:25 | of X in general is not equal to G F | |
19:27 | fx in general . But the special case exception to | |
19:30 | that is when we when we introduce the concept of | |
19:33 | an inverse function later , that's a very special function | |
19:36 | where the order of the composite actually doesn't matter . | |
19:39 | So we're gonna save that for later and you're gonna | |
19:41 | see how important inverse functions are in math . So | |
19:43 | solve these . Follow me on to the next lesson | |
19:45 | . Get more practice with composite functions right now . |
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Quality Math And Science Videos that feature step-by-step example problems!
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10 - What are Composite Functions? (Part 1) - Evaluating Composition of Functions & Examples is a free educational video by Math and Science.
This page not only allows students and teachers view 10 - What are Composite Functions? (Part 1) - Evaluating Composition of Functions & Examples videos but also find engaging Sample Questions, Apps, Pins, Worksheets, Books related to the following topics.