A Quick and Easy Functional Equation
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- Опубліковано 8 чер 2021
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For an equation of the kind f(g(x))=h(x) the solution is f(x)=h(g^-1(x))
That's neat!!! 😊
What guarantees that g has an inverse function?
The same way I solved this problem but it was longer than the method in video
@@yassinekh4382 ask arcsine
@@yassinekh4382 one one and onto guarantees
Note that a function of the form f(x) = (a x + b)/(c x + d) with a, b, c, d complex is a Möbius transformation, which is one of those things that crops up _absolutely everywhere_ . The technique of finding the roots (which doesn't help to solve this problem) can still be used to find the fixed points of the transformation, which can be incredibly useful (it's an automorphism).
Nice!
I had in mind, injective function, so we can solve x from y univoquely.
Take z=((2x-1)/(x-3)) and hence we get f(z) in terms of z. Then substitute z=x and we finally get f(x).
That’s precisely what I did
Exactly
y->x: good place to introduce the concept of bound variables and the constructs that use them. In this case the implicit lambda() in the function definition. Other familiar constructs that use them: summation, products, quantifiers, integrals.
I still not understand why you can replace y with x directly as you let (2x-1)/(x-3)=y? If you do this then x= (2x-1)/(x-3) ??
I think the reason this works is because the function is defined by a formula to compute any value in the range, it's not an equation with unknowns. For example a function defined by f(a)=a^2 can be also defined using f(x)=x^2, the variable chosen doesn't matter.
Absolutely correct!
Yes, I would even have written f(t) rather than f(y) or f(x). Only the function f matters.
Nice explanation on the replacement of variables...
(3x-1)^2/(x-2)^2 == f(x) the range of the function f(x) = [ 3 + 5/(x-2)]^2. So x2
I guess it's valid because you start with an *identity*, and are composing both sides of the equation with the function g(x)=x. It would work for any function where it's defined.
Yep but figuring where its defined can be tricky
I solved this correctly!
Yay!!! 😊
Congrats👍👏👏👏
I'm I'n class 9 I want to learn many things
@@MUJAHID96414 Same.
@@diogenissiganos5036 You are in class 9 and you solved this?
Trivial note: Solving (2x-1)/(x-3)=y can also be elegantly done using geometric analogy properties (2x-1)/[x-3 -(1/2)(2x-1)]=y/(1-y/2) etc
Functional equations are life....
Hehe we both did a functional equation video today
Really op!
functional equations - is our lifes
And mine is geometry 😎
That method you used in your video was very cool! 🤩
@@SyberMath to mathelite *
Excellent explained ❤️
I think the reason why you can replace f(y) with x is because x is just a parameter which passes in values. You just have to make sure that you replace all the y’s in the function with x.
Also there’s one thing I don’t quite get. You said that if you wanted to find the numerical value you do 2x-1/x-3 = x. But I don’t get why you would do this. Why are you allowed to equate that to x?
I’m new to this field and I have a question please.
If you replace x with (2x-1)/(x-3) and substitute in you get
F(x) =[ (2x-1)/(x-3) ] ^2 = (4x^2 + 4x+1) / (x^2- 6x +9)
So I have f(x) in terms of x. But it is not the solution in the video.
Clearly I’ve done something wrong . Can someone explain it to me please?
Thanks
What I can think is that when you equation x with (2x-1)/(x-3) and then you are finding f(x).But the problem is when equate both these you are indirectly stricted the domain of the f(x)(i.e.only two values which is the roots of the quadratic).So,for a real domain(or may be some elements less) it is good to equate to other variable like x',y,y' etc......In order to have a large domain...But your thinking is right when we have find a function having only two values in its domain
I HOPE THIS HELP
if You think something wrong please point me ...
Okay
Thanks for the video professor
Let t = (2x - 1)/(x - 3).
tx -3t = 2x - 1 or x = (3t - 1)/(t - 2).
f(t) = [(3t -1)^2](t - 2)^2.
That's exactly f.
Brilliant! Really like this one.
2x-1/x-3=2+5/(x-3)=t so x-3=5/t-2. then x=3t-1/t-2. f(t)=answer.
These are so much fun!
Beautiful solution.. Thanks for your effort.
Most welcome 😊
I really like how you did this video. First one I watched and I subscribed.
This is just stoking my addiction to Mathematics being almost 50 and did my Math degree about 30 years ago.
Anyway, great presentation on functions.
Awesome! Thank you!
This tells where the issue is on, which gets the question seen deeply.
I want to see his clip more.
5:30. Very important...What did we actually do? Glad you went into that
Thank you, honestly I don't 100% understand but I think by rewriting your tutorials I will be good in Maths. Still long way to go but I really want to be good in Maths like this.
Classic, one of your best, Sir
Thank you kindly! 🥰
This question is very easy at the beginning of the function 2x-1/x-3=A then you can continue
Thank you Sir for this great video.
since (2x-1)/(x-3) = 2 + 5/(x-3)
simply by this method
( 5/(y-2) +3 )^2
F(2x-1/x-3)=x^2=2 backdoor 2equation function. (1/2)^2 = 0
Yay! I did this on my own!
Good job! 😊
Sir can you please tell me what board application are you using. I want a good one but I have less knowledge regarding that. Please help me. Nice video btw.
Here in iran it's 1 a.m but I'm still watching.
Too easy for Chinese students
Simple and elegant ❤️
Thank you! 😊
How did you find the inverse of the function? I didn't understand the final part
if you put in the reverse function of (2x-1)/(x-3) (which is the constant in the denominator and the coefficient of x in the numerator getting switched & multiplied with (-) as a handy rule) as x in the function you will be left with f(x). it's really mind blowing. cannot recommend it enough if u r in hs!
Super Great, I have to whatch thid video many times, Thanks for sharing & teaching 👍🏻
Glad it was helpful!
Awesome trick at the end... 👍👍!!
Thanks ✌️
I think there's another way to solve this tho. Eliminate the -3 with x'+3 ish. Simplify isolating the 1/x type term. Substitute something like x'= 1/x ish or something like that, so x will now be alone with a number. Translation type substitution, finally. Reverse engineer for the domain.
Of course the way shown in the vid is better and faster, but you can avoid the whole variable talk since that is the way you do it from the get go.
Very nice explanation
The terms "independent variable" and "dependent variable" can clarify why to replace variables.
Please make a video on ' Strategies for solving Functional Equations ' , as you did for Diophantine Eqn...
Good idea. That would take a while to put together 😮😜
The problem was pretty simple and good explanation
Thank you! 😊
I love your share😆
nice job thank u sir
Most welcome
この問題は、写像と逆写像についての正しい理解が必要な良い問題ですね。解き方も、パズルのようでとても面白いです。
I'm japanese.
Hello!
I think that this is quick and easy , but this is nice ... This is genius 😃 ... I hope that this channel will the best math channel ) I thank you for your hard work and videos 💐 I wish you good luck )
Thanks a lot 😊💖
Apart from the domain. You cannot simply switch y back with x because they are not equal. y ≠ x, y = (2x-1)/(x-3)
It's a different x
@@SyberMath then it's pseudo scientific solution 😉
@@SyberMath Then we do not know the answer to the question, since the question is asking about a function of x...
Quick and easy but nice! ☺️
Thank you! 😊
@@SyberMath This video cleared all the problems about that other video. Thanks sir
@@debarghyaray2082 No problem!
Euler, your number is: e = 2,718
Great, But I didn't understand after minute 05:25 . . . Could you please explain in anothrr video how you found the inverse function, Thanks in advanced 🤝🏻
This is related to parametric equations.
Eliminating the parameter t gives the same equation.
that's so cool!
Thank you!
I literally watch your videos as im eating dinner.. got so much better at math thanks to u
It’s my pleasure! 🙂
Wow that was nice!
Glad to hear that ☺️
Super...👍👍👍
I solved it by using f-1(x^2)=2x-1/x-3 ==> f-1(x)=2sqrt(x)-1/sqrt(x)-3
Thank you for explaining/breaking the problem down in a way that makes sense.
Kennedy Torres - Professor Nell - Math2412
Hi Kennedy! Thanks for the kind words. 💖
I didn't understand "Professor Nell - Math2412"
@@SyberMathlike loading plugin
Hii
Hello
Why you can replace you with x directly?
Bu soruyu lys matematik hazırlanırken benzerini çözmüştüm güzel bir kısayol
you could replace y in original f(x).Result the same
Very good!
Glad you think so!
Đầu tiên tìm hàm số thỏa mãn. Sau đó tính giá trị. Cảm ơn nhiều.
hi What program you use for writinh
thanks very educational video
No problem. I use Notability
Thanks, I hope this will help me in entrance.
Np
You are so awesome 😎😎😎 your channel is growing fast!!!!
Thank you so much 😀
@@SyberMath could someone explain why is it that if f(x) is what has been found out, f((2x-1)/(x-3))does not work out to x^2? Am I making a mistake?
@@SyberMath thank u happy birthday 2 u for more than 1 year on UA-cam!!!!
GREAT
so good 👍
Thank you!
thank u for explaining
You're very welcome!!! 😊
Heartiest Greetings 💎🌙💕
I made x=(2t-1)/(t-3) and y=t^2, solved the x equation for t, then plugged the expression I got for t into the y equation.
But why your taken y in action bcoz it could be done with t and x itself
Hi man, what board do you use to make your videos? could you you me?
Notability
Thanks a lot for explanation
You are welcome
Can please tell me where are you from sir
2+5/(x-3)=y, so x= 5/(y-2)+3, f(x)=(5/(x-2)+3)^2
X^2 = f((2x-1)/(x-3)) = f( 2 + 5/(x-3)) = {[(2 + 5/(x-3)) -2 ]/5]^(-1) + 3}^2, therefore, f(y) ={ [(y-2)/5]^(-1) + 3}^2. Done.
I am a teacher in actuarial math (with industry background, not academia background). I am not a math guy (no degree in math at all). I just picked up math again after 45 years old. And recently I have brushed on this level of math in order to tutor my son as he is preparing his entry level actuarial exam(s). I solved this question in 5 minutes in my first try. I don’t see it is worth this long video. I am wondering I have missed the good points you try to express.
We do this type of shuffling and re-shuffling of writing of expression a lot in actuarial questions.
for x=3 why not error like in the first function?
Sweet
tricky but easy
please upload olmpiad,putnam or other competitive exam [JEE,NEET] questions
Sure! I will
I don’t agree with this solution. Replacing y with x at the end implies that y is equal to x. You defined y = (2x-1/x-3). In fact a given was that (2x-1/x-3) was NOT to be made equal to x.
x and y are dummy variables, you can replace them with anything
Yeah. Just look for the inverse from 2x-1/x-3 then subtituse x² with it
Even I did it.... OOH THAT SATISFACTION😊
i think that this problem must be defined with x≠2,right?
Im Japanese so i may not be able to answer right English
Very classic
Hi sir, what is the tool you use to write the text?
I use Notability
Nice
I still can't understand how can you change y into x when it is already one variable naming x
Shouldn't you add to the answer " & x!=3"
Nice!!!
Thanks!
This was a fun(ctional) exercise; thanks for sharing!
Hehe. That's punny 😜
@@SyberMath Thanks! Just let me know if you ever want to hear a pun on a particular topic.
so good.
Thanks!
Where did the x2 go? Why does that just disappear from all thought?
Bro, what are you using to write nicely? mouse?
Apple pencil
so cool
thanks
Good equation!
Glad you think so!
Klass Zor Uzbekistan👍
Thanks ...from 👍🇦🇷
Welcome!
why not write the domain of it
Fntstic easy to understand
関数方程式ですね
良い問題です
It is just a polynomial in a variable 'x'