so I always want to have some fun whenever I teach my math classes and let me give you guys an example
check this out this is actually the last question on my pre-calculus test recently and I think some
students were freaking out when they were doing this question and when they saw their answer
we'll see the question is we have a function 23 minus X over 1 plus 4X and our goal is to find
it's inverse function so this right here is a pretty standard pre-talkus equation and let me
just show you guys the pre-tocker sweat first we write f of x as y so we have y equals 23 minus
X over 1 plus 4X then we switch y to x x to Y so X is equal to 23 minus y over one plus four
why then we will have to isolate this y we can first multiply 1 plus 4y on both sides this way
this and that will cancel then distribute this so we get X plus 4 y well X Y like this that's equal
to 23 minus y now we will just move this to there and then move the X here so let me add the Y on
both sides like this and then minus the X like this this and that cancel this and not cancel
right here we have y plus 4 x y equals 23 minus X
both of them have the Y so we can factor out the Y and then we have y times 1 plus 4X equals 23
minus X and finally divide this on both sides we get y equals 23 minus X over 1 plus 4X and
this Y is exactly what we are trying to find namely F inverse and we have 23 minus X over
1 plus 4 x yay especially if this is last question I will feel good about this ready
um
does this look familiar do you feel like it's HIV right now
notice 23 minus X over 1 plus 4X is actually the same as the original
wow is this possible directly do anything wrong no we did that in fact this is possible
the inverse is actually the same as the original in this particular case
it's possible and whenever this happens this is the idea of the E for Lucian function so it's a
probability is not the best place to introduce this idea to the students the last question on the
test but I did give them a practice question that has similar flavor so if it did do the question
then they could expect this kind of things to happen so yeah and on that pre-calculus test that
I gave out recently I actually only use one two three four for the questions and also two zeros
so that's pretty good I do plan to make a live stream later on to go over the questions so
if you guys would like to know that uh leave a comment down below let me know
well let's discuss how I came up with this um a couple ways to do it one you can Wikipedia
this and you can see that there is a formula of word but that's the first way now let's
talk about the second way all right so here is the goal what we can do is the following
suppose we have a rational function because for the rational function you have to go
through all these computations I think it's a fair amount of work for pre-calculus students
so let's start with f of x and we want f of x to be a rational function in the following
sets I want a linear on the top over a linear on the bottom so ax plus b over CX plus d so that
and just like probably earlier the inverse is the same as the original so we want
f let's see that's this you don't have to right and store that every single time we want this
so that f of x is equal to its inverse but because we are trying to look for the inverse so let's go
ahead and write it on this we want the inverse of X we want the inverse of f I'm too happy to
talk about this right here all right we want the inverse of f to be the same as the original okay
yeah so how in the world can we do this well first we will have to assume that
this is invertible AKA one to one and all that I'll tell you it works and all that
so in this case this is how you can do it it's not the best way I will show you guys a second
way by the end of the video so you know watch out for that this right here it's the same as saying
we want F of f of x to be just X because you can apply the original F on both sides like this
right so that F and F inverse will cancel at X will be F of f of x so same thing like that
so now we have an equation to work out F of f of x that means we put f of x into itself so this is
f of x we want to enter that itself so how does that look like this means we have F of a X plus
b over c x plus d and we want this to be equal to X and let's see if we can fit in everything
in this board we'll put this into here and here so we want a times a X plus b over c x plus d plus b
over c times a X plus b over c x plus d and then plus d and we want this to be equal to X cool to
a certain degree because it's a lot of letters and our goal is to find ABCD ideally make them whole
numbers so that they look fairly simple and all that and I did tell you guys that I used one two
three four for this question as well all right so let's come back to here perhaps that's just
um multiply c x plus d on both sides so keep that in mind what we're doing we'll just
multiply c x plus d on both sides here right c x plus d on both sides not on both sides I
mean top and bottom right here CX plus d so when we take this time star they cancel and
let's distribute a little a so we are looking at a square X plus a b plus b times that so
it's C no what am I talking about B Converse B goes first before c so b c x and then plus b
D and then on the bottom this is not cancel distribute the little C so here we get this over
a c x and then plus b c plus this times that so it's C D X Plus D Squared and this is equal to
X and now multiply this on both sides so we get a square X plus a B plus b c x plus BX equals we
get a c x squared plus b c x plus c d x square plus d squared x this is not a derivative this
is just some number D squared times the X and you can move everything on one side
and all that up to you or you can now just equal coefficients so depends on how you want to do it
we have a polynomial and on the right hand side we also have a polynomial and um let me
just make sure that nothing is wrong something's wrong of course this is not BX this is B times d
B times D and of course this is B times d B times T and we are good now
so here we have x square and this is also x squared but on the left hand side we don't have
x squared term right so notice this notice that so from here because we don't have x squared term
that means AC plus c d uh let me just put this down here AC plus C D this has to be equal to zero
yeah and now if we look at the X term so here we have the a squared term a square X and this is
b c x oh bcx cancel on both sides why not let's just get rid of that bcx bcx gone
and we have this so you look so we have a square has to be equal to D Squared so a
squared has to be equal to D Squared yep and then what else do we have um
lastly here BD BD this is the constant third it has no X and on Randstad everything has X so
that means a b plus b d has to be equal to zero so here we have the three conditions from the
things that we have and uh if you look at this we can get rid of the C and then this implies
a is equal to negative d a is equal to negative D and right here it says a squared is equal to
D Squared but this is a better condition than that so we don't introduce this anymore right
because the square of that we have to make sure that the app is assigned and
now we have to look at B and D we know a is equal to negative D so you can say this
is a times let's write a as negative d right here and then we have this B and then plus b
D equals zero oh by the way I should have said c is not equal to zero here so that we have a
choice for C that's how we can carry off the C earlier because if C is equal to zero that
would be a little bit too boring so that's what I'll say that right here I will also say
if B is not equal to 0 then we can also get rid of that and because then this is also true so
it looks like we have a freedom to choose any value for C and likewise we have any
freedom to choose any value for B and that will make this true and that's now I have
to make sure a is equal to negative D then we are good to go so here is how
the equation will look like our function f of x will be equal to let me just replace
I like to use then then just replace negative into a so we have negative d
because a is equal to negative D and we have the X plus b B has the freedom because we got
rid of them earlier and then C is okay that's the freedom like this and then plus d like that cool
so if you look at this that's all good but if you want to get rid of the coefficients right here
that's also okay because we have this D here and this D here so we can divide everything by D and
again assuming D is not equal to zero right here let me just put it somewhere we want ABCD to not
be zero how's that just keep that in mind so this right here will give us Negative X plus b over D
over C over DX plus 1. and B over D we can just label that to be another constant likewise this
we can also label that to be another constant and this is how exactly Wikipedia so I will
just write it like Negative X Plus hmm beta well I'll just put it as capital B over this right here
is capital c x plus one so if f of x is in this particular form then you can have this F of f of
x is equal to X so this is the function that we want and c and b can be anything yeah C is equal
to this B is equal to that I think it's pretty cool and if you look at it I pick C2 B4 and then
we have the negative X right here and I pick B to be 23 so that's how I get that function
yeah but I will tell you though this right here is not the best way to talk about the involution
function because it's really computational however though I do think that if you want
to go crazier than maybe this is the way to go it took me a while right but here's the better one
check this out so another way to get a rational function that's Evolution we can do the following
I will start with f of x equal to K Over X and K is just any number don't say zero please
and the reason for that is because if we graph that then the picture is going to
be like this that's the case parsley for now then the picture will be for like this
right if K is greater than zero if K is less than zero then you have this and that instead
which is okay it's also symmetrical about Y equals X right so technically it should be more like this
and of course if you replace this with Y and you see that if you switch X and Y and
worked out the things that we did earlier you will see that the inverse is indeed the
same as the original but the idea is that this curve is symmetrical about this diagonal line
Y is equal to X that's why when you flip that we still get the original that's good
symmetrical about the diagonal line if we just move along the diagonal line
yes so we will start with f of x equals to K Over X and then we will move along was shipped along
the line Y is equal to X meaning we can go to the left however many you sorry we can go
to the right however many units and we just have to make sure we go up by the same unit
or we can also move to the left however many units and then go down however many units so if
this happens then we'll see this is going to give us a picture like this if I move it then I will
get to somewhere here and of course this will also move diagonally so it will be something like this
so this right here will still be symmetrical very good and of course we can also move down this way
so if we move this then that's pretty much it well how do we move the to the right however many units
and the same exact amount up so I can say uh right and up both let's say t units just on number t
right so how are we going to write that expression though well we start with f of x equals that so if
you want to move to the right T units that means we will have to have f of x minus t
this right here tells us we move the curve to the right T units and then we also have to go up T
units so make sure that we add the T afterward so this right here is exactly how we can move up and
down but to the right and to the up to the right and also upward T units and that's pretty much it
let's try expression for this and perhaps I will write it down this I will just say that
our new function G of x to be that f of x minus t plus t so G of X is equal to K over x minus t and
then we add T to it and right here of course we can just get the common denominator so x minus t
here and also x minus t here and we will see that this right here equals x minus t on the
bottom and then let's put down t plus T times x right here first and this is minus t Square
and then right here we add the K afterward so these are the X terms and these are the constant
terms this right here is better we use a so-called parameter this right here tells us how much we
have to move both to the right and also upward left and down as long as we are moving along this
line this right here is guaranteed that G inverse it's equal to original can just pick whatever
Theta you want plugging guarantee works as well so you can let me know which way you like more and I
know there are a lot more Evolution functions out there but I think this is perhaps my favorite one
and if you are a teacher as well you can also let me know what the questions that you'd like
to ask your students and feel free to use this for your students as well as always that's it