Can you do a video on how to solve functional equations in general or to solve different classes/types of functional equations please? Would be much appreicated as I find functional equations to be quite elegant.
@Bruno levi Levi &/- it does so only because . . . it’s an odd number of times fofof . . . of = x It wouldn’t work with an even number of times. Yes, that’s how it can be generalized. There’s another video with fof = x i.e. an even number of times . . . BUT that one works because it’s a product of terms with different powers, not the sum of terms, so the cancellation still works.
By the way, from the first equation, how would you determine x≠1, or is it purely written down because you know from the final solution that x cannot be 1 for f(x) to be defined?
Not doing this every day, I'm stumbling on why you are allowed to conveniently assign x=1/(1-x). Aren't you changing the parameters of the equation? If you do decide to do that, like a u-substitution for an integral, don't you have to change it back after you get it into form?
i think you will find that regardless of the initial value of x, cos(x) and sin(x) are in bounds(-1,1) , sin is montonic in the range, cos needs also to consider the value 0. so take two montonic ranges. then each successive application of the function narrows the range which i think converges on sin(x)=x and cos(x)=x those values are different and the ranges of answers for the two different functions do not overlap. You will need a calculator.
Damm! This is amazing! Also, how can you intuitively know to replace x with 1/(1-x)? Cause usually I will try special values and attempt to reduce the equation. Thanks :)
if you've practiced millions of maths problems, you will get that instinct; you know how to solve the problems. There are limited kinds of problems and limited ways of solving them, just try watch as many problem solving videos as you can.....
a bit too late, but i can answer you question: it`s not intuition, it`s logic. in functional equations, you are, basically, trying to get useful info. to achieve this, there are a few, let`s say, tricks. to know what tricks to use, you analyse the function. in this case, i analysed and can tell you that there are really few tricks one can use. in fact, there is only one: notice that f(x) is linked to f(1/1-x). therefore, if you know what f(1/1-x) is, you get your solution (that`s why he plugged f(1/1-x)). after this, you get f(1/1-x) is linked to f(x-1/x). applying the same logic, you want to know what f(x-1/x) is, so you plug it. then, you have f(x-1/x) is linked to f(x). do you see what happened? f(x) is linked to f(1/1-x), which is linked to f(x-1/x), which is linked to f(x). thus, you have f(x) linked to itself (the actual goal of a functional equation). now, you just solve the system of equation and get the value of f(x). btw, this is called the circle method.
@@caiodavi9829 Ahh I see, thanks for the detailed explanation! One thing I still don’t understand is that how you know the circular method works before even beginning to substitute. Like what is the insight behind x -> 1/1-x -> x-1/x -> x? Does it only work for 1/1-x or is there other fractions in the form p(x)/q(x) that works and why?
@@SyberMath WHY do that replacement why not plug in values for x like zero, one half and 2 and make equationa..you could do it that way..why wiuld anyone ever rhink of doing that replacement..i don't see why...
@@SyberMath Waitna minute you cant add the equations like that because you made a substitution..so what is f(x) in ome equation is not equals to f(x) in another and same with f(x-1/x)..see what I mean? So this can't be correct. When doing subsittution it's generally better to pick a different variable like u not the same one, as you know.
@@SyberMath can you respond to my comment above pretty sure you made a mistake when you added the two equations though..becsuse f of x for one equals f pf 1/1-× or whatever for the others..
At 4:12 equation 2 is written incorrectly first term should be 1/(1-x)
That's right
That's what actually confused me. Thanks for the correction david!
Author, please double check your video at the end for possible errors 😊
That really really made this confusing to watch :/ until I rewound. Should've looked at the comments I guess
-👌
Came here looking for an explanation regarding this as well. Thanks.
Can you do a video on how to solve functional equations in general or to solve different classes/types of functional equations please? Would be much appreicated as I find functional equations to be quite elegant.
They are elegant and that's a good idea!
@@SyberMath yes!
@@SyberMath have you done it I cannot see
@@SyberMath
Have you done it?
2:55 I didn’t get it how you simplify the expression
I love this. It's like magic out of Harry Potter.
"Specialis Revelio !"
Thank you!
Nice video !! I love how you solved it
I'm glad!
Functional equations are also good,pls put next video on differential equation
I feel like this can be solved only because of the fact that f(x) = 1/1-x kinda repeats, f(f(x)) = f^-1(x) and f(f(f(x))) = x
That is correct!
@Bruno levi Levi
&/- it does so only because . . . it’s an odd number of times fofof . . . of = x
It wouldn’t work with an even number of times.
Yes, that’s how it can be generalized.
There’s another video with fof = x i.e. an even number of times . . . BUT that one works because it’s a product of terms with different powers, not the sum of terms, so the cancellation still works.
There are many ways to solve different functional equations... Many times you have to play with it for some time.
By the way, from the first equation, how would you determine x≠1, or is it purely written down because you know from the final solution that x cannot be 1 for f(x) to be defined?
Good question. Because of the expression 1/(x-1), x cannot be 1
@@SyberMath Oh my god. How didn't I notice that?
It's literally in the thumbnail too.
At 3:43 the equation 2 is written wrongly, should be f( 1/(1-x) ) + f( (x-1)/x ) = 1/(1-x) . Please correct this or take video out!
No need to be so harsh. Besides, another commenter has commented and that comment is pinned.
thank you for this strategy I will surely use it!
Great!
Not doing this every day, I'm stumbling on why you are allowed to conveniently assign x=1/(1-x). Aren't you changing the parameters of the equation? If you do decide to do that, like a u-substitution for an integral, don't you have to change it back after you get it into form?
Wow ! The best of the best !
Thank you!!! 🥰
Thanks a lot ! Its a great channel.
Greetings from colombia 🙂
Thank you too! 😊
hope this channel get huge
💖
Matt Penn recently did this same functional equation on his channel.
Who is Matt Penn?
errors in f(1/(1-x))# f((1-x)/x) errors no simplfy
please more videos like this sir!
Sure!
I dunno how to do it and also dunno how did you come with that approach.. amazing 🤩
😊 thanks
Where can I get more practice problems on functional equations
www.amazon.com/Introduction-Functional-Equations-Problem-solving-Mathematical/dp/0821853147
www.amazon.com/Topics-Functional-Equations-Third-Xyz/dp/099934286X
www.math.uci.edu/~mathcircle/materials/M6L2.pdf
Great video as always
He doesn't generally do proof type problems but nevertheless it's a good question
Thank you! That's a good problem but a bit too hard for the channel, I think
It is
@@SyberMath IT IS WHAT?
i think you will find that regardless of the initial value of x, cos(x) and sin(x) are in bounds(-1,1) ,
sin is montonic in the range, cos needs also to consider the value 0. so take two montonic ranges.
then each successive application of the function narrows the range which i think converges on sin(x)=x and cos(x)=x those values are different and the ranges of answers for the two different functions do not overlap.
You will need a calculator.
Substituting 1/(1-x) instead of x will not affect the equation????.............
Damm! This is amazing! Also, how can you intuitively know to replace x with 1/(1-x)? Cause usually I will try special values and attempt to reduce the equation. Thanks :)
if you've practiced millions of maths problems, you will get that instinct; you know how to solve the problems. There are limited kinds of problems and limited ways of solving them, just try watch as many problem solving videos as you can.....
@@karljo8064 yes I've noticed this too many times. Experience is 👑
a bit too late, but i can answer you question: it`s not intuition, it`s logic. in functional equations, you are, basically, trying to get useful info. to achieve this, there are a few, let`s say, tricks. to know what tricks to use, you analyse the function.
in this case, i analysed and can tell you that there are really few tricks one can use. in fact, there is only one:
notice that f(x) is linked to f(1/1-x). therefore, if you know what f(1/1-x) is, you get your solution (that`s why he plugged f(1/1-x)). after this, you get f(1/1-x) is linked to f(x-1/x). applying the same logic, you want to know what f(x-1/x) is, so you plug it. then, you have f(x-1/x) is linked to f(x).
do you see what happened? f(x) is linked to f(1/1-x), which is linked to f(x-1/x), which is linked to f(x). thus, you have f(x) linked to itself (the actual goal of a functional equation). now, you just solve the system of equation and get the value of f(x). btw, this is called the circle method.
@@caiodavi9829 Ahh I see, thanks for the detailed explanation! One thing I still don’t understand is that how you know the circular method works before even beginning to substitute. Like what is the insight behind x -> 1/1-x -> x-1/x -> x? Does it only work for 1/1-x or is there other fractions in the form p(x)/q(x) that works and why?
Hi professor. Very nice.do you have chanel on the Instagram?
une erreure a 4:32 eq 2
yes
I like this ❤️
Bazı videolarda yorumlarda türkçe yazmışsın türkiyeden misin yoksa çeviri mi kullandın ?
Siz turkler cok soru soruyor 😂
@@SyberMath o kadar yanıt verince ve aksanından ben de türk olduğunu sandım :D
4：07第2式寫錯了
típico problema de olimpiadas de matemáticas, así como para soltar la mano jaja ... saludos desde Chile bro
Saludos
I guess this works because a certain matrix has order 3 in PSL(3,Z).
Excelent I enjoyed the video
Awesome, thank you!
Is SyberMath Turkish by any chance?
Isso é Bom demais!
f(1-x/x) minus f(1/1-x) is zero is a great lesson for me my maths professor (my own sister) gives me zero mark and says idiot!!!!!!
😂
There are errors
Problem with this solve
f(super) + f(1/(1-super)) = super
😀😀😀
👍👍👍
🙃😊
Hi SyberMath...may you solve the functional equation...
f(x)+f(1/x)=1
Thank you for attention ::
If f(x) = 1/2, it works!
I get why x!=1 but why x !=0?
because of (x-1)/x
@@SyberMath Thanks :D
No es correcto.
f((1-x)/x) no es igual, en principio a f(1/(1-x)).
It should better be good. Bc it's both rational and functional.
Is this high school level?
More like competition/olympiad level
These steps would fail in characteristic 2.
Pas facile les chose besoin une axiome spledide
Functional equations are good
Absolutely!
@@SyberMath WHY do that replacement why not plug in values for x like zero, one half and 2 and make equationa..you could do it that way..why wiuld anyone ever rhink of doing that replacement..i don't see why...
@@SyberMath Waitna minute you cant add the equations like that because you made a substitution..so what is f(x) in ome equation is not equals to f(x) in another and same with f(x-1/x)..see what I mean? So this can't be correct. When doing subsittution it's generally better to pick a different variable like u not the same one, as you know.
It's like taking the composition of two functions. For example f(2x+1) is the same as f(g(x)) where g(x)=2x+1
@@SyberMath can you respond to my comment above pretty sure you made a mistake when you added the two equations though..becsuse f of x for one equals f pf 1/1-× or whatever for the others..
nice problem...
Thanks
I think you are fast on algebraic simplifications and I'm sorry to tell that it's not purely error-free sometimes.
Thank you! I know. I make mistakes
Are you turkish
❤
Madagascar
Reminds me the movie 😁
good
Thanks
L'ho fatto tempo fa e il risultato è.... f(x) =(1/2)(x^3-x+1)/(x-1)x se ben ricordo
👍
classic
Thanks!
You are losing 90% of your audience by not using another variable.
Copied from michael penn bruh
Anyways,nice video
I did not copy from Michael Penn bruh
😂
@@SyberMath oh sorry, he did the same problem a few months ago though
Likeee
Take a hearcut and get yourself a job
😂