Learn AASL 2.2.3 Inverse functions in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 2.2—Functions, notation domain, range and inverse as reflection.
Learn AASL 2.2.3 Inverse functions in this free IB Mathematics Analysis and Approaches (AA) video lesson for SL 2.2—Functions, notation domain, range and inverse as reflection.
The video discusses inverse functions, explaining that they essentially reverse the mapping of a function. For a function that maps a domain to a range , the inverse function maps back to , effectively swapping the domain and range.
Key points include:
Overall, understanding the relationship between the domain and range of a function and its inverse is crucial for sketching and evaluating these functions effectively.
00:00This is not going to
00:01look at inverse functions. Now,
00:04inverse functions basically what it
00:06means is when you have
00:08your domain and your inputs
00:10and you get outputs, this
00:12is your domain, this is
00:13your range, your function goes
00:15from A to B. This
00:16was A and this is
00:17B, A to B. Well,
00:19simply the inverse function, or
00:21F minus one as it's
00:22called, goes backwards, it brings
00:26
00:28This is f minus one
00:29and then the range of
00:31the function becomes the domain
00:33of the inverse function and
00:34the range of the inverse
00:35function is the domain of
00:37the function. So they swap.
00:41Again, like to use my
00:43analogy with the Coke machine,
00:46imagine I put in coins,
00:48press a button and I
00:49get out of kind of
00:50Diet Coke. Well, the inverse
00:52function is, it's my
00:56It's like a refund machine.
00:57I'm now going to put
00:58in my kind of Coke.
00:59There's a function, a function
01:00is going to do something.
01:02And I get that money,
01:05my refund. Now, these only
01:08work for one to one
01:09functions. And the reason is
01:11because if my domain, if
01:15the range of my function
01:17is now the domain of
01:18my inverse function, well, if
01:19this was a many to
01:20one, what would happen here
01:22is the inverse function
01:24become a one to many,
01:26and we know a one
01:27to many isn't a function.
01:29So it has to be
01:32a one to one function
01:33in order to have an
01:35inverse. If it's not a
01:36one to one, it won't
01:38have an inverse function. Simple
01:39as that. Okay, first we're
01:43going to look at sketching
01:44an inverse function, and then
01:45we're actually going to try
01:46and evaluate some in first
01:47functions. So what happens is,
01:52The x's that I put
01:55into the function give me
01:58y's. This is my normal
02:00function. This is, should probably
02:02write here, this is y
02:05equals f of x. This
02:07is my f function. I
02:09put in x's and I
02:10get out y's. Now the
02:12inverse function basically means those
02:15y's that I got out,
02:16I'm now gonna put them
02:18into the function. They're gonna
02:19be my new x's.
02:20and I'm gonna get out
02:22new wise which will actually
02:23be the x's of the
02:25original function. Now that, as
02:26I say, it sounds complicated,
02:30but it basically means everything
02:32gets switched. So see this
02:33point here, and I'm gonna
02:34actually, let's look at, I
02:36want to look at the,
02:38I wanna look at the
02:39specific points that where it's
02:40actually right on a point.
02:42This isn't negative four, negative
02:43two. Let's look at this
02:44one here, zero, negative one.
02:46This is two, zero.
02:48This is 3, 3, and
02:52this is 4, negative 5.
02:54Now what actually happens is
02:56the negative 4, negative 2,
02:59because everything gets flipped, becomes
03:02a negative 2, negative 4.
03:04So it actually becomes here.
03:05So this is negative 4,
03:07negative 2. For the inverse
03:08function, it becomes negative 2,
03:10negative 4, because it's flipped.
03:13The y's are now the
03:14x's and the x's are
03:15now the y's.
03:16This zero negative one becomes
03:20negative one zero. This two
03:23zero becomes zero two. This
03:26three three actually just stays
03:28as three three and this
03:29four five becomes five four.
03:33Now maybe you can see
03:35what's actually happened here but
03:37it ends up being a
03:40reflection in the line y
03:42equals x which is this
03:44diagonal
03:44So this line is y
03:48equals x. So whenever you're
03:49sketching an inverse function, I
03:52want you to draw that
03:52line. Draw the line y
03:54equals x. You'd actually get
03:55a mark for recognizing that
04:00it is a reflection in
04:03this line. Now, I want
04:06to join these dots. Now,
04:08not the ones on the
04:09red line, but these ones,
04:10I want to join these
04:12is not very with me
04:14as I try to join
04:15them. I'm going to join
04:16this here to there. It's
04:23actually going to meet there
04:25and like this, it's a
04:26bit like a bit like
04:27a sweet. So you can
04:29see here all these points
04:31it's a perfect reflection on
04:33that line. You can also
04:34see the coordinates of, well,
04:38the intercepts of the axes
04:39where the function
04:40and f intercepts at zero
04:43negative one, sorry, it intercepts
04:46the y -axis at negative
04:47one, the inverse function intercepts
04:50the x -axis at negative
04:50one, and where the f
04:52intercepts the x -axis at
04:55two, the inverse function intercepts
04:56the y -axis at two.
04:57So this is actually y,
04:59it was f negative one
05:01of x. So this is
05:02a very standard question you're
05:04gonna get asked to actually
05:05just sketch the inverse function.
05:06I'm gonna do an example,
05:08town.
05:08Here. Now before we actually
05:12sketch it, I want us
05:13to be able to evaluate
05:17some of these functions. So
05:18here it gives me the
05:19domain. This is the domain
05:22of f, negative 4 to
05:235. And you can see
05:24it goes from negative 4
05:25to 5. I've already done
05:29a lesson on evaluating functions,
05:30but let's do one of
05:31these anyway. f of negative
05:334 means I go to
05:34negative 4. Down here is
05:36the graph.
05:36What's the y value negative
05:383? Easy. But this is
05:42the new one now that
05:43I haven't come across. This
05:45is f minus 1 of
05:471. Now what that actually
05:48means is it's like saying,
05:52well first it's like saying
05:53when I put 1 into
05:54the inverse function what comes
05:55out. But if I don't
05:56know the inverse function, if
05:57I only have the function,
05:59it's like saying 1 has
06:00now come out. What did
06:02I put in? Or another
06:03way I might write it
06:04is
06:04f of what equals one.
06:10So the way we do
06:11this is, again, see if
06:13you can figure it out
06:14before I tell you a
06:15press pause in the video.
06:18F minus one of one
06:19is I go to where
06:21I go to one as
06:23the output for F. So
06:24the y value of one,
06:26I go to one and
06:27it's here and the graph
06:28is actually right there. Then
06:30I go down, what is
06:31it? Zero.
06:33That's one of one is
06:34zero. Let me just remove
06:35this. F minus one of
06:38one is zero. Let's try
06:41F minus one of three.
06:43So now I'm gonna go
06:43to where Y is three
06:45here. I go find the
06:47graph, where is it? It's
06:48over here. Find the graph.
06:50Go down, it's five. So
06:51F minus one of three
06:52is five. Or F of
06:54five equals three. You can
06:56see it there. And finally,
06:57I've just done another F
06:59of negative three.
07:01this negative one. Go to
07:03negative three and down to
07:04negative one. So when you're
07:06trying to do these together,
07:08it can become confusing. This
07:11one you go when it's
07:12a normal f of something,
07:14you just go to the
07:16x value and then find
07:17the y, but when it's
07:18a negative one, you go
07:20to the y value and
07:22find the x. Okay, hope
07:24that makes sense. I am
07:26going to now find the
07:28range of f and the
07:29range
07:29of f minus 1, and
07:30then we're going to sketch
07:30it. So the range of
07:32f, that's pretty straightforward. It
07:34goes from negative 3 to
07:373. The range of f,
07:40and it's going to be,
07:41sorry, it's this one, the
07:42range of f is y
07:43goes from negative 3 to
07:483. I'm including negative 3
07:50and 3 because he's including
07:51negative 4 and 5. This
07:52is the domain of f.
07:53And this is the range
07:54of f because it goes
07:55from here, it goes from
07:57negative
07:57negative 3, up to negative
07:593. Let's arrange. Find the
08:03range of f minus 1.
08:04Now you could sketch it
08:05first and then figure it
08:07out from that. But it's
08:09actually, I can actually, I
08:12can figure it out from
08:13this because I know, because
08:16I know the domain of
08:18f, the domain of f
08:19is negative 1, negative 4
08:20to 5. I know the
08:22domain of f is the
08:24same as the
08:25of f minus 1. So
08:27the range of this is
08:28the same as this except
08:30obviously it's y. So y
08:31goes from negative 4 to
08:355. Now once I sketch
08:37it, you'll see that makes
08:39a bit more sense. So
08:40now I want to sketch
08:41the graph of f minus
08:411. I want to sketch
08:42it on this graph. The
08:43whole element is always as
08:45you just sketch it on
08:47the same grid. So the
08:50first thing I do always
08:52is draw this line
08:53draw the line y equals
08:55x, use a ruler, straight
08:58line, there. And then I'm
09:02going to start picking, obviously
09:04you don't have to use
09:05a different color, but I'm
09:06going to do it. I'm
09:06going to start picking the
09:07points that I know that
09:09are on, that are right
09:11on a particular point. So
09:13here, this is negative 4,
09:14negative 3. I'm going to
09:15now, that becomes negative 3,
09:17negative 4. Or just, if
09:19I go straight towards this
09:21line,
09:21particular half a box here,
09:23half a box here. This
09:26is negative three, negative one.
09:28Sorry, you need to go
09:28to negative one, negative three.
09:31Or a full box over,
09:32a full box over. Let's
09:35go to the next, what's
09:36my next line? Okay, what
09:37he crosses the, well, he
09:40crosses the, he crosses the
09:42x axis at negative one
09:43points something. So he's going
09:44to cross the y axis
09:45at negative one points at
09:46a nugget exactly where it
09:47is. Maybe let's go here
09:48something like that.
09:49He's crossing the y -axis
09:54at one, so this guy's
09:55going to cross the x
09:56-axis at one. Now this
10:00is important. They are going
10:01to meet, they meet right
10:03there. They meet on the
10:05line y equals x, because
10:07it's reflected in the line
10:08y equals x. It's itself.
10:10It's reflected onto itself. That's
10:11just right there. And then
10:13finally, this point, which is
10:155, 3, is going
10:17going to become three, five.
10:19Now I've actually not included
10:21that point. And but it's,
10:24let's imagine it's there somewhere.
10:26Okay, and now I just
10:27need to join these dots.
10:29This is a bit I
10:30always don't like, especially as
10:32at the end of a
10:33lesson. Because it would mean
10:35I have to bend the
10:36entire lesson if I don't
10:37do it well. And that's
10:40going to have to do.
10:41So you can see it
10:41goes through all these different
10:43points. And yeah, you would
10:44get full marks
10:45for that, you can clearly
10:47see is your reflection in
10:49the line y equals x.
10:50So just to recap, we
10:52need to know how the
10:54domain and range relate for
10:55the function and the inverse
10:56function. The domain of the
10:59function is the same as
11:00the range of the inverse
11:00function and the range of
11:02the function is the same
11:03as the domain of the
11:05inverse function. First of that,
11:06correct? We also need to
11:08know how to evaluate the
11:10functions and the inverse functions
11:11and we need to know
11:13how to sketch the inverse
11:16function. And that's it.