I Solved A Homemade Functional Equation in Two Ways
Вставка
- Опубліковано 20 лис 2023
- 🤩 Hello everyone, I'm very excited to bring you a new channel (aplusbi)
Enjoy...and thank you for your support!!! 🧡🥰🎉🥳🧡
/ @sybermathshorts
/ @aplusbi
⭐ Join this channel to get access to perks:→ bit.ly/3cBgfR1
My merch → teespring.com/stores/sybermat...
Follow me → / sybermath
Subscribe → ua-cam.com/users/SyberMath?sub...
⭐ Suggest → forms.gle/A5bGhTyZqYw937W58
If you need to post a picture of your solution or idea:
intent/tweet?text...
#functions #functionalequations #algebra
via @UA-cam @Apple @Desmos @NotabilityApp @googledocs @canva
An Exponential Equation | 2^{3^{5^x}} = 3^{2^{5^x}}: • An Exponential Equatio...
A Surprising Exponential Equation: • A Surprising Exponenti...
PLAYLISTS 🎵 :
Number Theory Problems: • Number Theory Problems
Challenging Math Problems: • Challenging Math Problems
Trigonometry Problems: • Trigonometry Problems
Diophantine Equations and Systems: • Diophantine Equations ...
Calculus: • Calculus
Bruh just plug it in
That's what I was thinking of
😂😂😂😂😂
Seriously man, bruh moment
3
Seriously that was so easy
That 2nd method is so simple it feels like cheating!
The second method was pretty obvious.
Actually there is no need for f(x) to be linear; it could be anything. For instance, f(x) = √(x + 1) and then g(x) = x² − x − 1.
Q: Solve this functional equation. f(x)+f(1/x)=1, for x=/=0, x=/=1. The craziest part, is that I know of a solution, which is a rational function in x, but Idk how you would generally solve this problem. SyberMath has made videos on similar problems, but this one in particular I am stuck.
@@pokemonjourneysfan5925 There are two restrictions: f(1) = 1/2 and some function of x which gives the negative value if applied to x ^ -1. This implies a logarithmic function so f(x) = 1/2 + C * ln(x).
@@MrGeorge1896 Not exactly. And second, I assumed the function is a rational function of the form (ax+b)/(cx+d) for constants a, b, c and d.
@@pokemonjourneysfan5925 Yeah, you are right, such solutions exist. I found two different solutions:
f(x) = (ax + b) / [(a - b) (x - 1)] with a ≠ b and not both a = 0 and b = 0
f(x) = (ax + b) / [(a + b) (x + 1)] with a ≠ -b and not both a = 0 and b = 0
Bruh...just put x=3 in the question..seriously that was easy...
Seriously. How is this a 9-minute video?
uh... 3 ? less than 5 seconds
Very well thought out 🥂🥂
Thank you!
Estupendo!
3
3
since nothing is known about g(x), its impossible to solve this problem. for all we know f(x) might not be have multiple values for some x, which im pretty sure makes it incorrectly defined.
we do have some info on g(x)
you're right, i didnt specify my complaint correctly. What i see as problematic, is that we dont specify anywhere that g(x) + x is injective, which would mean that defining the f function in this way might cause some problems. I do realise that this is somewhat petty for lack of a better word, but i believe that it's a important thing to state.
Sybermath: I have a challenge:
Here is a system of equations:
g(f(x+a) - f(x)) = x
g(f(x)) -x/(x+1)g(f(x+1)) = 0
Find what f(x) and g(x) is (class of solutions is okay as well) Bros please bump this comment!
Answer: 3
Surely just plug in x=3 as that's the value given.
The question should be revised to make that we can find the c. Problem will be more interesting. 😉😉😉😉😉😉
3
i have a doubt , can the second method be considered valid in math exams where problems are subjective and u needto give steps
In general, it should be ok but of course it depends on the professor (based on how picky and annoying they are)
i see tysm sir @@SyberMath
x=3 g(x)+x=5+3=8 so f(8)=3
genau, so habe ich auch gedacht !!
f( 8 ) = f ( g(3) + 3) = 3
We know g(3)=5, so how does x=3 play out:
f(g(3)+3)=3
f(5+3)=3
f(8)=3. We have the answer to the question.
I don't know why this video is so long. No assumption of linearity required.
but he use a different ways
bro just x=3
So another interesting take could be to solve for f and g, but without the linear assumption.
1] Use method 2 in the video to get f(8) = 3 like before.
2] Given this, prove that g must be a constant function. This can be done by creating a very simple system of equations. Using the knowledge that f(8) = 3, we know that when g(x) + x = 8 then x = 3. Subtracting these gives us that g(x) = 5 for all x.
3] By plugging in 5 for g(x) into the original equation we get another functional equation, f(5 + x) = x.
4] Setting x = t - 5 the above equation becomes f(5 + t - 5) = t - 5, which simplifies to f(t) = t - 5.
5] Finally, change of variables gives us f(x) = x - 5
thats incorrect.
we dont get g(x) = 5 for all x. we only know that g(x) = 5 when x=3.
What you did is exuivilant(pardon spelling) to saying that since x+2+x=8 when x=3, then x+2 =5 for all x.
Are you kidding me? Just put x=3 on both sides of the equation and obtain: f(8)=3
sure thing! no kidding 😁
Method 1: Long, complicated, uses an assumption that may or may not be true(f can be any invertible function)
Method 2: Just plug in dummies😜