10 - What are Composite Functions? (Part 1) - Evaluating Composition of Functions & Examples - Free Educational videos for Students in K-12

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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