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
00:26the B back to A.
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.