12 - What are Inverse Functions? (Part 1) - Find the Inverse of a Function & Graph - Free Educational videos for Students in K-12

12 - What are Inverse Functions? (Part 1) - Find the Inverse of a Function & Graph - By Math and Science

Transcript


00:00	Hello . Welcome back . The title of this lesson
00:02	is called inverse functions . This is part one of
00:05	several . So here's a concept that gives a lot
00:08	of students a lot of problems . And even when
00:10	you think you understand it , there are usually a
00:13	few little aspects that you really ever either didn't think
00:15	about or really just didn't understand . To begin with
00:17	. What I want to do is make sure you
00:19	understand by the end of this lesson what an inverse
00:21	function is , what the graph of an inverse function
00:24	really looks like , but mostly just intuitively to know
00:27	what an inverse is and why it's important in math
00:30	. All right . So let me let you take
00:32	a trip down memory lane with me , just for
00:34	a second , we know that we understand how to
00:37	solve equations . Generally in order to solve an equation
00:40	for X , we have to kind of undo everything
00:43	that's happening to X to put X by itself on
00:46	one side of the equal sign . If something is
00:48	added to one side , and then we have going
00:49	to subtract to get X by itself , if something
00:52	is multiplied by X . And we usually divide ,
00:54	we do the opposite . Up until now we've been
00:57	saying that we've been doing the opposite to undo the
01:00	operations to get X by itself . Really , we're
01:02	going to dive a little bit more into that .
01:04	And we're going to talk about the idea that you've
01:06	already been kind of using inverse is up until now
01:09	the additive inverse you've been using , that's what we
01:11	call to undo addition and the multiplication , or the
01:15	multiplying inverse . That's what we do to undo multiplication
01:18	. So we've already been doing that , but now
01:19	we're putting different words , were calling them in verses
01:22	, right ? But then we're going to extend that
01:24	concept to functions , functions can have inverse is also
01:28	and that's where the wheels come off the train a
01:31	little bit for a lot of students , because they
01:33	don't they can kind of understand what it is ,
01:34	but they have no idea why we're doing it .
01:36	The reason that we're doing it is because later down
01:38	the road , when we solve more complicated equations in
01:41	order to get X by itself , when we're solving
01:43	equations , we're going to do the opposite , do
01:46	the undo the undo operation . In other words ,
01:49	we'll apply the inverse function in order to get X
01:52	by itself to solve more complicated equations . Right now
01:55	you've been doing in verses a lot , even subtracting
01:57	and multiplying or dividing whatever . But later down the
02:00	road , especially with exponential functions , we're going to
02:03	have to do a inverse function to get X by
02:05	itself . And so now we have to learn what
02:07	it in verses . And then we're gonna apply that
02:09	as we start to to learn what the inverse of
02:11	the exponential function is . And by the way ,
02:13	it's called a logarithms . We're gonna talk about logarithms
02:16	. So this is very important to understand what the
02:18	algorithm is . So , first let's go down and
02:21	make sure you understand what some of these other things
02:24	are . So we have the concept of the additive
02:28	in verse . And believe me , I know a
02:32	lot of people probably want to skip through this because
02:34	they know what addition is . But just trust me
02:36	, go through it with me because the conclusions I'm
02:39	going to draw about additive inverse is directly related to
02:42	what an inverse function is . So , it's very
02:44	important that you follow me through this . So let's
02:48	say you start with a number any number you want
02:50	? I'm gonna pick a number 14 , that seems
02:52	like a nice number . All right . And I'm
02:54	going to add something to it . So , I'm
02:55	gonna change this number . I'm gonna add , you
02:57	know , let's change colors . I'm gonna add something
02:59	to 14 and change that number . I'm going to
03:02	add five to it . So now if I were
03:04	to add that together , that would be 19 .
03:06	But let's say I want to undo this , I
03:08	want to change it back into 14 . Well ,
03:10	I have to do the opposite of the addition that
03:12	I did here , which would be subtraction of the
03:15	same number five . And what I would get out
03:17	as a result of that would be 14 . Exactly
03:20	what I started with . Okay , you might say
03:23	this is pretty simple and pretty dumb . Why is
03:25	he teaching me this ? What I'm trying to say
03:27	here is the additive in verse of The # five
03:37	is the number -5 . Now , you see that
03:40	symmetry of the number line . The positive numbers and
03:43	the negative numbers . The negative numbers are the additive
03:44	inverse is of the positive numbers . You probably saw
03:47	this definition way back in Algebra one , but you
03:50	don't care back then because who cares ? They're additive
03:53	inverse . Great . What you've done here is you
03:55	take a number , You change it by adding something
03:57	to it . You have to do the inverse to
03:59	undo that to get back what you started with .
04:02	I want you to remember this concept of getting back
04:04	what you started with because this concept of getting back
04:07	what you started with is exactly how we're going to
04:09	tackle inverse functions here in just a second . All
04:12	right . So , as another example , let's say
04:15	you start with 20 the number 20 . I'm just
04:17	making this number up . And let's say you subtract
04:20	six from 20 but then you want to undo that
04:22	. How do you wanna do that ? Where you
04:23	have to add six back ? What do you get
04:25	back As a result 20 ? You start with something
04:28	, you do something to it , you immediately undo
04:30	it and you get back exactly what you started with
04:32	. Right ? And so you can say here that
04:36	um the additive in verse of negative six is Positive
04:45	six . Okay , so we talked about additive inverse
04:48	is awesome . How do we apply that ? Well
04:51	, let's talk about something related to that . Let's
04:53	talk about the multiplication in verse . It's a really
05:01	similar concept . It's not any harder at all .
05:04	All right . Let's pick a number out of thin
05:06	air . Can be any number I want . Let's
05:07	pick 16 . And let's say I multiply 16 by
05:11	something . Let's say I multiply it by four ,
05:13	but I want to get back 16 . So ,
05:15	how do I undo this multiplication ? I can't add
05:17	or subtract the negative four . That's not going to
05:19	undo that because this is multiplication . How do I
05:22	get the 16 back ? Well , I'm gonna have
05:24	to multiply by 1/4 Right ? And that's gonna give
05:28	me 16 . So you see by multiplying by four
05:31	, I've changed the original number , but I can
05:33	immediately undo that by multiplying by the inverse of Ford
05:36	. But not the additive inverse . It's the multiplication
05:39	inverse . So what it's telling you is that the
05:42	inverse of multiplication is division because you're multiplying something .
05:45	But what you're multiplying by is division . So the
05:47	inverse of multiplication division . The inverse of addition is
05:51	subtraction . Is exactly what I said in words at
05:53	the very beginning of this lesson . The goal is
05:56	if you start with something , you multiply by something
05:58	, you multiply by the inverse to undo that you
06:00	get back . What you started with . We get
06:02	back what we started with we get back what we
06:04	started with . That's what I want you to remember
06:05	all throughout this lesson . Because inverse functions are all
06:09	about how to get back what you started with .
06:11	So we say That this multiplication in verse 1/4 .
06:15	We say that it un does it um does it's
06:18	not a technical word , that's my word button does
06:20	the multiplication by the four . So they kind of
06:22	annihilate each other and you end up with what you
06:24	basically started with . So now we're gonna move the
06:27	train along a little bit and talk about inverse functions
06:30	. Because that's where again students start to get ,
06:32	they have no idea why we're doing this or anything
06:34	. But basically what happens is an inverse function is
06:37	what we want to kind of undo what the first
06:40	function does . It's like the opposite function . Think
06:42	of peanut butter and jelly , or mayonnaise , and
06:45	mustard , or ketchup , and mustard . Whatever you
06:47	have these little pairs of things that go together .
06:48	Every function I shouldn't say every function . Every function
06:51	that we will talk . We'll talk about how to
06:54	figure out if the function has its inverse . But
06:56	most functions have a cousin function called an inverse function
07:00	that essentially un does what the original function is doing
07:04	. And so mathematically , that means it's going to
07:06	literally take an undo the mathematical machinery of that first
07:10	function with the inverse function mechanic . So , let's
07:14	take a concrete example . All right , let's see
07:18	how much space do I have ? I don't want
07:19	to save some room . Let's say I have a
07:21	function F of X . And it's equal to X
07:25	plus four . Over to This is a really simple
07:28	function . And let's say I have another function G
07:30	. Of X and it's equal to two x minus
07:34	four . Now , I'm gonna give you the punch
07:36	line , these are inverse functions of one another .
07:38	You cannot tell by looking at them that their inverse
07:41	. So just forget it , especially if I give
07:42	you more complex function , you'll never be able to
07:44	look at it and just say , oh yeah ,
07:46	those are inverse function . You're never gonna be able
07:47	to do that . I'll show you how to be
07:49	able to tell what their inverse functions , but I'll
07:51	tell you ahead of time these are inverse functions .
07:53	So they're special . They go together right there very
07:56	uh they have a lot of synergy between them ,
07:58	you could say , right , let's take a look
08:00	at what that might look like , what that synergy
08:02	is . Let's take and look at the following thing
08:06	. Yeah , let's look at G . Of one
08:11	. This is the function G let's put the number
08:12	one into their , what would we get ? We
08:14	put the number one into their , this is the
08:16	input , right ? We put the input into here
08:18	. What do we get ? two times 1 -4
08:21	. That's what we get . So then when we
08:23	calculate that we're gonna get to then minus four ,
08:26	we'll get negative to . So G F one is
08:28	equal to negative two . Now , just like we
08:32	learned in the last section with composite functions , you
08:34	can change these functions together or nest them however you
08:37	want to think about it . We talked about composite
08:39	functions in the last lesson . If you haven't done
08:41	that , you have to know that to get this
08:43	. So let's take the output of this function uh
08:46	here . And let's run it through the other function
08:48	which you know ahead of time . It's an inverse
08:50	function . So let's see what happens . Take this
08:52	guy and let's run it through the F . Function
08:54	, which is the cousin function of that guy .
08:56	And what we get is F . Of G .
08:58	Of one , which means we've done this . So
09:01	then we say , let's run the number negative two
09:04	in through the F function , because that's what we
09:06	got is an output . We stick it into the
09:09	outermost function here . Okay , What do we get
09:11	? Take this over here . What we're going to
09:13	get is what I want to do it . Let's
09:16	do it right here . It's gonna be negative two
09:18	plus four over to just take it and stick it
09:21	into here . What do we get here ? Negative
09:22	two plus four is going to be 2/2 . So
09:26	what we get is one , so F of G
09:29	. Of one Is equal to one . I want
09:33	to make sure you understand what's happened here and and
09:35	in fact it's not so obvious at first . But
09:38	look back to what we did in the beginning .
09:40	I said additive inverse . We start with some number
09:42	, we add something to it , but we do
09:44	the inverse and we get back what we started ,
09:46	we start with a number , we subtract something ,
09:48	we do the inverse . We get the number back
09:50	. The inverse gives you back what you started with
09:52	it and does the thing that you've done to it
09:54	. So here we multiply by four . We do
09:55	the universe and we get back where we started here
09:58	. We know that these are inverse functions because I'm
10:01	telling you they are . If you put the number
10:02	one in here , we get this out . But
10:04	if we take and run that through the other function
10:06	, we get the number one out . We put
10:09	a one in and we get on one out .
10:11	That's exactly what we did there . We start with
10:13	a number , we do something , we do the
10:15	universe and we get what we started with . We
10:17	start with the number we do a function , we
10:19	do its inverse and we get the number out inverse
10:22	functions undo each other . I'll say it again .
10:26	Inverse functions undo each other . I'll say it a
10:29	third time . An inverse function un does what the
10:32	other function does . That's what they are there special
10:34	functions . These are not random functions off , you
10:37	know , just randomly taken their specially crafted to be
10:40	in verses of one another . Right ? So let's
10:44	spell this out a little bit more . Okay ,
10:47	I'll say note , what do we do here ?
10:50	G . Of one . That's what we started with
10:52	. We calculated that we got an answer of negative
10:55	two . But then we ran that function through or
10:59	the answer through the other inverse function which is F
11:04	. And we got an answer of one out .
11:06	We started the chain by putting a one in .
11:08	We went through this calculation intermediate value than running it
11:12	through the inverse . And we get a number out
11:13	that's exactly equal to what we started with . So
11:16	these are the same . These are the same .
11:21	Now I want to go off to the side here
11:25	and show you something . We ran . We did
11:29	uh F . Of G . Of one . Now
11:31	let's go over here to the side . And this
11:33	calculate something similar instead of F . Of G .
11:35	Of one . Let's calculate G . Of F .
11:40	Of one . So this is a composite function .
11:44	We gotta one . We're gonna do the composite function
11:46	again . But we're gonna flip it around . And
11:48	if you remember I told you in the last lesson
11:50	when we did composite functions , I said in general
11:52	when you flip the order of the composite function ,
11:54	you do not get the same thing . But there's
11:57	a big exception and the big exception are inverse functions
12:00	, inverse functions always undo each other no matter what
12:04	order you do them , that's why they're special .
12:06	So here we did F . Of G . Of
12:07	one here . Let's do G . Of F .
12:09	Of one . So how do we do that ?
12:12	Well we say , well f . of one is
12:15	this it's going to be one plus 4/2 Which is
12:19	5/2 . It's an ugly fraction . But let's take
12:23	that ugly fraction and we're gonna put it into here
12:25	. So g . of f . of one means
12:28	that what we do have a five halves here .
12:30	So we do G . Of five halves because that's
12:33	what we calculated . We're feeding that into the G
12:35	function . What is the G function ? It's this
12:38	it's two times the input here , which is five
12:42	halves minus four . So I've taken this and I'm
12:45	sticking it into the G function . But notice the
12:48	two's cancel . So what you get is G .
12:50	Of F . Of one is the two's cancel .
12:54	So you have just a five left over minus four
12:56	so G of F . Of one . It's just
12:59	equal to one . And notice what happens . Is
13:03	that this is the same thing . Mhm . All
13:09	right . That's the that's one of the major conclusions
13:12	I want you to remember from this . I told
13:14	you a great paint in great detail in the last
13:16	lesson . When you flip the order of the composite
13:19	function , you're not gonna get the same thing .
13:20	But that's for random functions that pull a random function
13:23	F of X . Is X squared , pull another
13:25	random function F of X is two X -3 .
13:28	Okay , those are random . If I do the
13:30	composite functions in both orders , in the different orders
13:34	, I'm gonna get different answers . But inverse functions
13:37	are special . They always undo each other . I
13:39	put the input in of a one . I run
13:42	it through one of the functions . Then I immediately
13:44	run it through the inverse function . Then I get
13:46	the same exact number out . I stick a number
13:48	in here . I run it through the G function
13:50	, I run it through its inverse which is the
13:52	F function . I get exactly the same number out
13:54	which is one . All right . And that's going
13:58	to be the case of all inverse function pairs .
14:00	Alright , So make sure I haven't missed anything so
14:02	far . They undo each other G NF undo each
14:05	other . And you're always going to get that input
14:07	function , that input number back out . So what
14:09	I want to do now is I want to give
14:11	you a few more examples of this whole undoing business
14:15	and then after that I want to um graph the
14:19	inverse function . So let's rewrite these functions here just
14:21	to have them handy . Ffx is X plus 4/2
14:27	. G . Fx two . X -4 . Same
14:32	functions . I haven't changed anything . I just want
14:34	to have uh everything down low here . Now let's
14:37	do the same thing . Let's say F of negative
14:40	three . Let's put a negative three in here .
14:43	So the negative three plus 4/2 . So we add
14:48	them on the top and we get a one and
14:50	we get so we get a one half out of
14:52	that . So what we figured out is F of
14:54	-3 . This intermediate answer is one half after we
14:56	run it through one of the functions . Let's take
14:59	that and then run it through the other functions .
15:01	So you'll say that G of F of negative three
15:06	Is going to be G of 1/2 is going to
15:11	be equal to , I have to put in one
15:12	half in year two times one half for x -4
15:16	. The 2s are going to cancel . So what
15:19	I'm going to have here is just the tools are
15:20	gonna cancel . So we're gonna have a 1 -4
15:22	which is three . Notice the three is exactly what
15:26	I'm sorry ? Not a 3 -3 . One minus
15:29	four is negative three . It's exactly what I started
15:31	with exactly my input here . So G of F
15:34	of negative three is surprised negative three . Because these
15:38	are inverse functions . Whatever number I put in ,
15:42	I run it through one of the functions and I
15:44	run it through the other function and then I get
15:46	exactly what I started with . It's exactly like undoing
15:48	each other from before . So just to spell it
15:51	out one more time , I could say , I
15:55	could say uh Let's see here f of -3 Yields
16:01	a value of 1/2 . But then I put that
16:04	through the other function G of one half , and
16:07	that yields a function of . Sorry here . Yeah
16:11	. Do you have one half yields a number of
16:13	negative three ? So I start the chain by putting
16:15	a value of negative three in and I get out
16:17	of the chain or out of the nest . However
16:19	you want to look at it the same exact input
16:21	value . They've undone each other . It turns out
16:24	that this undoing of one another is going to help
16:26	us solve lots of equations down the way that we
16:29	don't know how to solve now . Mostly in the
16:31	beginning , will be solving exponential equations because the exponential
16:34	function has an inverse . I'm going to talk about
16:36	it later . It's called the law algorithm . So
16:38	if you take the law algorithm of both sides of
16:40	the equation , you kind of undo the exponential function
16:44	and then bam the variable drops out because you've kind
16:46	of annihilated and eliminated the exponential part . Because you've
16:50	done it's inverse , you've undone it just like we
16:52	add in order to undo subtraction and we divide two
16:56	under division , we do the inverse , which might
16:58	be taking the law algorithm to undo the exponential function
17:01	here , we don't have any exponential . I'm just
17:03	explaining what these things are , what these inverse functions
17:07	are . All right . So then let's do one
17:12	more just to bring it home here . We have
17:16	said with these two functions , we put a number
17:18	in of one . Run it through both of these
17:21	functions and we get a number one out . We
17:23	reverse the order of the functions . Doesn't matter .
17:26	We still stick a number one in . We get
17:27	a number one out . Let's change the input .
17:30	We put a number negative three and we run it
17:31	through both functions . We get a negative three out
17:33	. Now , let's not put a number in .
17:36	Let's just put a variable in . Let me rewrite
17:38	the functions F of X is equal to X plus
17:42	4/2 , and G of X Is two X -4
17:50	. Same functions . Now , instead of putting a
17:52	number in calculating it and putting into another and calculating
17:55	it . Let's just do this . Let's calculate generally
17:58	F of G of X . In other words ,
18:01	we're not putting a number and we're leaving the input
18:04	of that innermost function , Just a variable . We're
18:07	letting the input to the whole process be any number
18:10	we want we just call it X . So when
18:12	we put X in two G of X , what
18:14	do we get ? We get this and we have
18:15	to take this thing and put it into the F
18:18	location . So that means we take this whole thing
18:20	and stick it where X is . So it'll be
18:22	two x minus four plus 4/2 . We just took
18:27	G of X , whatever it was and we stick
18:29	it in the X location right there . Now look
18:31	at what you have , you can drop the parentheses
18:33	now two x minus four plus four . Over to
18:37	this is going to go to zero . So you're
18:39	gonna have to X over two which is gonna give
18:42	you once you cancel X notice what's happened , We
18:45	stick a value of X . M Which means any
18:47	value I want . I run it through both functions
18:50	and I get the same exact value back . That's
18:52	exactly what was happening over there . Put a one
18:55	in , get a one out , put a negative
18:57	three in . Get a negative three out . Put
18:59	any value I want in for X . I'd stick
19:00	it exactly the same value of X out . That's
19:02	what that's telling you . When you operate an inverse
19:05	on uh the inverse function on the other function .
19:08	They annihilate each other and you're just left with whatever
19:10	input you had to the whole chain . Yeah .
19:14	All right . Now , just to make sure we're
19:16	on the same page , let's run that same process
19:18	through in reverse instead of F of G of X
19:20	. Let's do G of F of X . Again
19:24	leaving it general we have ffx , we're gonna put
19:26	it inside of G . We have to stick this
19:27	whole thing and put it in there . So to
19:30	be too times X . But this X . Is
19:33	the entire function X plus 4/2 . And then we
19:38	have to uh do this guy now when we have
19:41	a two on the outside and one half here they
19:43	can cancel like this . And so what you're going
19:46	to have is X plus four left over . That's
19:49	all that's left out of this . But then you
19:51	have a minus four and look here , this goes
19:53	to zero . So you get an X out so
19:56	you stick an X in , you run it through
19:57	both functions in reverse order from before and we still
20:01	get exactly the same thing . So what we have
20:03	concluded is that F of G of X is equal
20:09	to X . And we've concluded that G of F
20:13	of X is equal to X . Now , I
20:18	could have started this whole lesson by giving you this
20:21	gibberish here and said , hey , there are these
20:23	things called inverse functions F of G of X .
20:26	X M G of F of X is X .
20:28	Don't you understand ? Nobody's going to understand that .
20:31	I don't even understand that when I say it out
20:33	loud doesn't make any sense . But by going through
20:35	the whole thing , I hope you can understand what
20:36	this means . It's this is what you would typically
20:40	see in most textbooks as the definition of the inverse
20:43	function . And that definition goes like this F and
20:49	G . R in verse functions if the following thing
21:00	, if this is true , right ? So this
21:04	is what I'm gonna circle as the definition of the
21:06	inverse function . So , somewhere in your algebra book
21:08	or pre calculus book or calculus book or whatever ,
21:11	they're gonna define this thing called an inverse function .
21:13	They're gonna say F and G are inverse functions .
21:15	If F of G of X , X , N
21:18	G of F of X is X . And then
21:20	probably have some other words about the domain . You
21:22	have to make sure the domain this is true in
21:24	the domain of G and that this is true of
21:26	the domain of F . But assuming both of these
21:28	are smooth continuous functions with no , you know ,
21:31	assume taub's infinities or anything , they're very well behaved
21:33	functions , then this is going to be true for
21:35	all X . For all input values of X .
21:38	Whatever input you put in , you're gonna get an
21:40	input , you're gonna get exactly the same number coming
21:43	out of the other end . Once you run them
21:44	through both of these functions , it won't matter the
21:47	order and what you do it . This is the
21:49	definition of an inverse function that you're gonna see in
21:51	most books . But we just went through the process
21:53	here so that you can hopefully understand it a little
21:55	bit more . All right . Now , what I
21:57	want to do is just as important as everything we've
22:00	done up to this point . We want to graph
22:02	these functions . I want to show you what a
22:04	graph of a function looks like right next to what
22:07	the graph of its inverse looks like , because it
22:10	turns out there's a really easy way to visualize what
22:13	an inverse function uh will look like . And it
22:16	helps us understand what they're what they're really doing .
22:18	All right . What an interest functions really doing .
22:20	So , what we wanna do is want to plot
22:22	both of these functions . All right . So ,
22:25	we have these two functions here . We have F
22:26	of X is equal to X plus four . Over
22:30	to That was function number one , and G fx
22:34	Is two , X -4 . This is a line
22:38	and this you might not realize it , but it's
22:40	also a line . Let's kind of manipulate this a
22:42	little bit f of X . This can be broken
22:46	up as X over two plus four over to just
22:49	divide each term by two . Of course . This
22:51	is going to be F of X one half times
22:55	X plus two Mx plus B . Mx plus B
23:00	. They're different lines . The Y intercepts are different
23:03	and the slopes are different but they're both lines .
23:06	So you can immediately tell that they're both going to
23:08	do something but they're not gonna have any squiggles and
23:11	they're not gonna have any parameters . Are not gonna
23:12	have anything crazy like a lips or circle or anything
23:15	like that . They're both can look like lines .
23:17	So what I wanna do is I wanna graph this
23:19	f . Function . I want to graph this right
23:23	on the same graph as this guy . So it's
23:25	probably gonna take a little bit of time . But
23:27	I really encourage you to hang on with me because
23:29	it's really important um for us to get there .
23:33	It's very important for us to actually um to do
23:37	that to do this together . So what I want
23:39	to do is , first of all right , the
23:40	functions down again , right ? The first function let
23:44	go and write him up in the upper left hand
23:46	corner here . The first function F . Of X
23:50	. Was one half X plus two . That's what
23:52	we just wrote down . And the other function G
23:55	. Of X . Is two X -4 like this
24:00	. So , those are the two functions we want
24:01	to graph , fortunately they're really easy to graph because
24:04	they're just lines . So what do I want to
24:06	use to graph these guys ? What color ? Let's
24:08	try this one . That's probably not gonna look so
24:10	great . Let's try . Mm We'll try this one
24:15	. Okay , the first one . All right .
24:17	We have for the first one . Why intercept of
24:20	two ? So here's one . Here's two . So
24:21	I'm gonna put a dot right here . It passes
24:23	through this point . But the slope on this one
24:26	is one half X . So that means I rise
24:28	one and I run to okay rise one or 12
24:33	There's different ways in which you can do it .
24:35	But let's go and do it that way up 1/1
24:38	, 2 . Because the scale is one tick mark
24:41	and the scales one tick marks all up and then
24:43	over to like this and then I can go uh
24:48	I can do the same thing here . I can
24:49	go Down one and over to like this , something
24:54	like this . Okay , because it's going to be
24:57	up 1/2 , up one over to something like that
25:00	. So I can draw a line smoothly through those
25:02	points . So let's do that real quick and see
25:05	if I can not drop my paper in the process
25:07	here . Try to do the best I can .
25:09	It's not going to be perfect , but let's try
25:12	to get it as close as we can because there
25:15	actually is something really neat that I want you to
25:17	be able to see . Yeah , so it's going
25:21	to look something like this like that . All right
25:27	, let's just double check myself . Okay , so
25:29	this function F of X is equal to the one
25:34	half X plus two . But remember that just came
25:36	from X plus 4/2 . It was the same exact
25:39	thing . So we'll say X plus 4/2 . So
25:43	this is the graph of this function . All right
25:47	. Um Then let's go and grab the other one
25:53	and for that one I think I want to try
25:54	to use let's use black . So two X minus
25:58	four . So we have minus four means a Y
26:00	intercept negative one , negative two negative three negative four
26:03	crosses right there and a slope of two . That
26:05	means up to over one , up to over one
26:09	like this . Up to over one like this .
26:12	So it goes through those three points . Let me
26:15	see if I can graph it through there . Something
26:21	like this . I can get there . It's gonna
26:26	look something like this . Okay , something like that
26:36	. Not perfect . You get the idea now in
26:38	your mind I want you to extend this blue line
26:41	. Of course it goes on and on forever and
26:42	extend the black line that goes on and on forever
26:45	. Now , the interesting thing I want to show
26:47	you is , well , let me go ahead and
26:48	draw , guess write this down . This is um
26:52	f of X is two X -4 . That's what
26:56	that is now . Don't they look interesting ? It
26:59	looks like they're a mirror image of one another .
27:02	One of the functions , its inverse is just a
27:05	reflection of that function over a certain line . Can
27:07	you spot what that line is ? The reflection of
27:11	one to the other is going to be through this
27:14	diagonal line right here . So let me take this
27:17	off and try to draw this line . So this
27:22	is where the mirror the mirror would lie if we
27:25	were gonna actually flip this guy over here . So
27:28	to get the inverse , all you do is you
27:30	take the function graphically and you flip it over this
27:34	red dot in line . And the red dotted line
27:36	is a diagonal line , Y is equal to X
27:39	. It's straight down the middle , it's Y .
27:41	Is equal to X . Because the intercept zero and
27:43	the slope here is one rise one run , one
27:47	rise run rise one . So it's just a dotted
27:50	line of a diagonal of 45 degrees , which means
27:52	it goes right and it splits these guys into so
27:55	graphically , you never do this graphically . But if
27:57	I just said , hey , here's a function ,
27:59	right ? It's a line graph , it's inverse .
28:02	All you would do is somehow mentally map all of
28:05	these points across the dotted line here and then draw
28:08	your line . And that would be the inverse function
28:09	if the blue line , we're not an actual line
28:12	but some other function . Because we're gonna have in
28:14	verses of all kinds of functions , alright , then
28:17	the same thing will hold if this if this um
28:20	if this thing had some kind of squiggle in it
28:22	, then we could still reflected over and we would
28:24	have a different squiggly line , but it would be
28:26	reflected over to the other side . That would be
28:28	its inverse . Now , I want to talk about
28:31	a couple of things , I want to mark a
28:33	few really important points off of this graph . Okay
28:38	, let's see what is this point right here ,
28:40	this is negative two comma one . This point right
28:42	here is negative two comma one . How can I
28:46	read that ? Negative two comma one . If I
28:47	run it through this function right here , if I
28:49	put negative two into here then I'm gonna get a
28:51	negative one . I'm going to add it here ,
28:53	I'm gonna get a one . So negative two comma
28:54	one . What other point is on this guy ?
28:56	Let's go to -3 . And then what is this
28:59	? This intersection right here ? So at this point
29:02	is negative three comma one half . You can just
29:06	read it right off the graph . Here's one and
29:08	here's one half there . If you put a negative
29:10	three in here , you do the math here ,
29:12	you're gonna get one half out . All right .
29:14	So , these points are also on the inverse function
29:18	over here . But the way it works is what
29:21	happens is these points are going to be exactly flipped
29:24	around . So let's go and take a look at
29:25	that by switching to black . And let's say ,
29:28	what would this point B the mirror image . In
29:30	fact , you can see I already have it here
29:31	. The mirror image when you reflect it over .
29:33	Is this point so negative two comma one . This
29:36	point becomes one comma negative too . Look at that
29:41	and stare at it . What you've done is the
29:44	inverse maps every point on the function to another point
29:48	. That is kind of its mirror image cousin .
29:49	But what happens is I take the coordinates and I
29:52	flip them around . So negative two comma one becomes
29:54	one comma negative too . And let's look at this
29:58	. If I've taken literally go straight through this guy
30:00	and down here , it's going to be this point
30:02	right here , right ? Which is negative three comma
30:07	I'm sorry , not negative three comma It's going to
30:09	be 1/2 common negative 3 . 1 half , comment
30:14	negative three . This is one half and then common
30:16	negative three . It's right there . So this point
30:18	maps with this one and this point maps with this
30:20	one . Every point on this line is going to
30:22	have a cousin partner point on the inverse line .
30:25	And every point here is going to relate to every
30:27	point here by simply taking the coordinates and flipping them
30:30	backwards , taking the coordinates and flipping them backwards .
30:34	And that flipping backwards business of taking every point on
30:38	the function and flipping them backwards to make the inverse
30:42	. That is why the functions undo each other .
30:44	Right ? Because if you think about it , if
30:46	this is the inverse , sorry , if this is
30:48	the function and this is its inverse , then if
30:51	I take any number in , let's say I put
30:53	negative two as my input , I put negative two
30:56	in . I get a one out but then I
30:58	take that output and I stick it into the other
31:01	function . I put a one in And what am
31:03	I going to get a negative two out ? You
31:05	can see I stick to take the output of this
31:07	, stick it in here and I'm going to read
31:09	off a negative two out . So I started with
31:12	negative two . I ran it through this function .
31:14	I run the output into the other function and then
31:17	I get the same exact thing back , started with
31:19	negative two And I ended with -2 . Right same
31:24	thing here . If I stick as a starting point
31:26	negative three in , I get a one half out
31:28	. If I take that answer and put a one
31:30	half into the other functions , I get a negative
31:32	three out . So you see that's what we were
31:34	doing all along here . I put a negative three
31:37	in . Put it through both of the functions .
31:38	I get a negative three out . I get exactly
31:40	what I started with . Put a one in ,
31:42	put it through both functions and get a one out
31:44	. Put a negative three in . Run it through
31:47	, take this point , run it through . I
31:48	get exactly what I started with . Start with a
31:50	negative to get the intermediate . Run it through here
31:52	. I get the negative two out . So graphically
31:55	I'm showing you why this undoing business works . It's
31:59	because this thing is a mirror image reflection over 45
32:02	degrees , which basically means every point here has to
32:05	be flipped around like this , which means every time
32:08	I stick it in , put in here and get
32:09	a number out and then take that number here and
32:11	get the corresponding number out . I'm going to get
32:13	exactly what I started with . So the bottom line
32:17	punchline most important thing I want you to get out
32:19	of this other than the math that we've done before
32:22	Is if a function has an inverse , if a
32:26	function has an inverse , then that inverse will be
32:29	that function reflected over a 45° line . Why is
32:33	equal to X . This is the inverse function of
32:35	this and the same thing goes true . If this
32:38	is your input function then it's inverse will be the
32:40	blue line . So it's not that It's not one
32:43	way there in verses of each other . F is
32:45	an inverse of G . & G is an inverse
32:48	of f . And they're both mirror image reflections obviously
32:51	over that line . 45° with the points flipping around
32:54	. And that is why they undo each other .
32:57	All right . So if you can , in your
32:58	mind envision this being more complicated than a line and
33:01	you can flip it over , you would still have
33:03	the curve but it would be rotated down the same
33:05	kind of thing would hold . All right . Um
33:09	One more thing I want to talk about before we
33:11	close because I've really pretty much gotten everything out that
33:14	I want to get out . But one important thing
33:16	I want to say before I get into go off
33:19	to the to the very final end is that not
33:22	all functions have in verses . Most of them do
33:26	, but not all of them . I want to
33:27	tell you really quickly how you can tell if a
33:30	function has an inverse and we're gonna practice it more
33:31	in the next lesson . Alright , for an F
33:34	to pass for a function to have an inverse .
33:37	It must pass what we call a horizontal line test
33:40	basically . So four F of X to have an
33:45	inverse . It must pass the horizontal line test .
33:53	I'm not gonna write that out . Horizontal line test
33:55	is very , very easy to understand . All right
33:57	, let's take a look at this first of all
33:59	. Let's look at what we have here . A
34:00	horizontal line test means if you have a function and
34:04	you start crossing a horizontal line through this function ,
34:08	it can only cross in one spot , horizontal line
34:10	only cuts this function in one location anywhere . So
34:13	that means when I map it then the inverse function
34:16	will pass the vertical line test . Because for a
34:19	function to be a function it has to pass the
34:21	vertical line test . So if you're going to say
34:23	this thing has an inverse function , you you need
34:26	to make sure it passes a horizontal line test so
34:28	that when you flip it it'll pass a vertical line
34:30	test to be a function . So let's take a
34:34	look at something that doesn't have an inverse . Right
34:36	. As an example , let's say we have a
34:38	graph here and we have our nice friendly parabola ffx
34:44	, his ex square . Does this function have an
34:46	inverse over this whole domain like this ? Well ,
34:48	all you have to do is say , well does
34:50	it pass a horizontal line test ? Well no fail
34:55	the horizontal line test because it cuts into locations .
35:00	If it cuts in more than one location then uh
35:03	it's not gonna work now . What would happen if
35:05	you actually tried to construct an inverse with this thing
35:08	? Right ? What would happen ? Remember ? The
35:09	inverse function would just be the mirror image reflection of
35:12	this thing over that 45 degree line right here .
35:15	So what would happen is the inverse function would be
35:18	something like this because if you remember that 45 degree
35:21	line is somewhere somewhere like this . So the original
35:25	function was like this . You flip it over .
35:27	The inverse would be something like this . But the
35:30	inverse has to be a function . And remember all
35:33	functions have to pass the vertical line test . This
35:36	thing fails the vertical line test because it cuts in
35:39	more than one location . So the reason this thing
35:42	doesn't have an inverse , the reason we have a
35:43	horizontal line test is because once you mirror image reflected
35:47	it has to pass a vertical line test . So
35:49	if it has to pass , if it fails a
35:51	vertical line test and it's going to fail a horizontal
35:53	line test for the original function . So in most
35:56	books , what you see is they'll tell you ,
35:58	you know , if a function has an inverse ,
36:00	it has to pass a horizontal line test . All
36:04	lines are gonna pass or except for horizontal lines are
36:07	going to pass horizontal line test , right ? So
36:10	all lines have an inverse . But this parabola over
36:14	this entire domain like this does not have an inverse
36:16	. So you that's what you would write down on
36:17	your test . But other functions that we could draw
36:19	will have an inverse . And in the next lesson
36:21	we're gonna get a little bit more practice with that
36:24	. So we have learned a lot in this lesson
36:27	. It's a really important topic because inverse functions feed
36:30	into so many of the more advanced kind of uh
36:34	lessons in the curriculum . Whatever your classes studying ,
36:36	pre calculus , calculus , or algebra , inverse functions
36:39	feed into that . We talked about an additive inverse
36:42	. All we're doing is when we add something ,
36:44	we have an inverse to undo it , we get
36:46	the same number back , We get the same number
36:49	back with multiplication by essentially multiplying by a fraction or
36:52	division , uh to get the same number back .
36:55	And then we said , functions can have inverse is
36:57	also these are inverse functions . Now we know why
37:00	with their graphs because their reflection across the 45° line
37:04	like that , but basically , whatever number we put
37:06	into this thing , we're going to get it back
37:08	once we run it through both of the functions .
37:10	And we prove that is true by relaxing the numbers
37:13	and just letting the input be a variable . You
37:15	run it through , you get the same number out
37:17	that you put in , you flip the order .
37:18	You run it through , you get the same number
37:20	out that you put it in . So inverse functions
37:23	are inverse , is if you run it through both
37:26	of the functions and get the same thing out ,
37:28	no matter the order in which you Uh execute that
37:31	guy . Then we plotted both of those functions and
37:34	showed that the line in this case of the function
37:38	has an inverse by reflecting it over a 45° line
37:41	like this . And we said that every point on
37:43	the original function has a corresponding point on the inverse
37:46	function where the coordinates are flipped around . And the
37:48	reason the inverse functions undo each other is because of
37:52	those coordinates flipping around . So if I stick a
37:54	number into the original function , I get an intermediate
37:57	number out . I take that number into the other
37:59	function and I get the same number in that eye
38:02	out that I stuck in to begin with . And
38:04	that will happen for any number that you put into
38:07	this inverse function pair . It's a lot of material
38:11	. But make sure you understand this concept because in
38:13	the next lesson we're going to be calculating the inverse
38:16	function , figuring out what the inverse . I just
38:19	told you these are inverse funds . I didn't tell
38:20	you how to figure it out . I just said
38:21	here they are . In the next lesson , we
38:23	want to actually calculate what the inverse function is and
38:27	sketch them and figure out if it passes the horizontal
38:29	line test and all of that stuff . So make
38:31	sure you understand this . Follow me on to the
38:32	next lesson will continue working with inverse functions in algebra
38:37	, pre calculus and calculus .

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