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

## 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
02:42 what an inverse function is . So , it's very
02:44 important that you follow me through this . So let's
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
04:17 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
04:28 , 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
04:48 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
05:58 , 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
08:39 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 .
09:42 , we add something to it , but we do
09:44 the inverse and we get back what we started ,
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 .
10:13 a number , we do something , we do the
10:15 universe and we get what we started with . 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
31:50 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
36:42 . 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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